We should first establish what we mean by the Assyrian and Persian Empires, since those names have been used to describe some very different cultures over the course of about 4,000 years. For the purposes of comparison, it seems reasonable to select the Neo-Assyrian Empire which began with Adad-nirari II in 911 BC and the Achaemenid Empire of Persia founded by Cyrus the Great in 550 BC, since these are the closest in time. Both these reigns began with extensive new conquests which raised the empire in question to the status of a great power. The Neo-Assyrian Empire fell in 609 BC and eventually became a province of the Achaemenid Empire. The Achaemenids lasted until 330 BC, when Alexander the Great captured Persepolis.

Although the Neo-Assyrians lasted longer than the Achaemenids, their rule was less secure and was maintained by terror. The massive fortifications of Nineveh were there for a reason: the conquering Assyrians ruled their subjects very harshly and were widely hated. The Persians were more tolerant and were able to rule a much larger empire. Their ceremonial capital at Persepolis required no fortress around it. Although the Achaemenids were in most respects much more civilized rulers than the Assyrians, they had nothing to equal the great library of Ashurbanipal, one of the largest and most important repositories of texts in the ancient world, which contained over 30,000 cuneiform tablets. This library, however, was only amassed in the declining years of the empire, by its last great king.

The Assyrians and Persians both ruled in roughly the same place, namely, Mesopotamia; however, they ruled in very different ways. The Assyrians were mainly a military power and practiced particularly effective military tactics such as digging beneath a city's walls to weaken them before attempting to conquer the people within the city walls. The Assyrians conquered people and collected tribute. They kept conquered people in check by using brutality, including slavery. The Assyrian armies were renowned for their brutal tactics. The Assyrians did not allow conquered people to govern themselves and punished them mercilessly if they did not provide tribute. They were so hated that other people celebrated when their capital, Nineveh, was eventually destroyed.

The Persians, on the other hand, ruled over an organized empire with a benevolent form of government. Once they conquered another territory, they ruled with tolerance for other cultures and traditions. For example, the Persian leader Cyrus did not allow his armies to burn and loot conquered cities, and he allowed Jews to return to Jerusalem, their capital. The Persians placed local governors called "satraps" in positions of power and subdued conquered people by offering them peace and an efficient system of government. 

The Assyrian Empire and the Persian Empire were two of the earliest major empires in the world.  The Assyrians were powerful from about 900 to about 600 BC.  The Persian Empire came afterwards, holding power beginning around 550 BC.  Both were empires in what we now call the Middle East.  Both were monarchies.  However, there were very important differences between the two.

A good way to express these differences is to say that, from our perspective, the Persian Empire was more enlightened.  Because of this, it was possible for them to rule a much larger empire than the Assyrians could.  The Assyrians had a much more autocratic and centralized system of government.  The king of Assyria was seen as all-powerful.  Everyone else in their society was seen as a slave of the king.  When the Assyrians conquered other peoples, they tried to dominate them and exploit them.  The Assyrians essentially used the outer parts of their kingdom as colonies to be exploited, not as valued parts of their own society.

By contrast, the Persian Empire was more enlightened.  The king did not portray himself as a completely dominant figure.  The people of the empire were portrayed as strong and important members of society.  The empire was not exploited as ruthlessly for the benefit of the center.  Instead, each part of the empire had its own governor and was seen as an equal part of the empire.  Tribute was demanded from each region, but not as ruthlessly as in the Assyrian Empire.

Thus, the Assyrians and the Persians were both empires, but they ran in rather different ways. 

What similarities exist between the Assyrian and Persian Empires?

The Assyrian and Persian Empires were both large, multicultural, and multi-ethnic empires of the ancient world. The Assyrians rose to dominance first, collapsing by the end of the seventh century BCE. Their empire would, at the its height, extend across Mesopotamia and the Middle East as far west as Egypt. Persia's greatest period of expansion, on the other hand, would come under Cyrus the Great (who would die around 530 BCE). The Persian Empire was much larger than the Assyrian Empire (and history's largest empire up to that time), conquering much of the same territory that the Assyrians had, but extending further west across Asia Minor and further east towards the Indus River.

Both the Persians and Assyrians were monarchies, which ruled with the assistance of elaborate systems of administration (to more effectively govern these far-flung, multi-cultural empires). As historian D. Brendan Neagle writes about the Assyrians:

The lands subdued by his campaigns were turned into provinces administered from Assyria by a governor directly responsible to the king. In Assyria itself existing administrative divisions were reduced in size to improve central control. An efficient communication system was set up using messengers who carried reports back and forth from the governors to the central administration (The Ancient World: A Social and Cultural History (5th Ed). Upper Saddle River, New Jersey: Prentice Hall, 2002, p. 63).

When looking at this description, you can certainly see parallels between the Assyrian and Persian systems of administration. Later, the Persians would subdivide their empire into the various satrapies, while also relying heavily on internal communications, tying their empire together by the use of road-building.

Additionally, both empires had a great deal of internal instability, which rulers had to reckon with. To again quote Neagle,

When Babylon revolted in 689 BC, Sennacherib (704-681 BC) destroyed it....In 671 B.C. Egypt was subdued but quickly rebelled and had to be beaten down again eight years later. On the death of Ashurbanipal in 627 BC, a dynastic struggle racked the Assyrian state. Babylon again revolted, and under the Aramaean (or Chaldaean) Nabopolassar (625–605 BC) made common cause with the Medes of Iran and bands of Scythian nomads from Eurasia...In the west Palestine was again in revolt. In 614 BC Assur fell, and two years later Nineveh was captured. By 609 BC the remnants of the Assyrian army were being stamped out in northern Mesopotamia (65)

Similarly, the Persians also had to reckon with a significant degree of internal instability. Among the most historically famous revolts was the Ionian Revolt by the Greek cities in Asia Minor, which famously precipitated the Persian Wars. As a second example, you can point towards Egypt, which successfully drove out the Persians in 404 BCE (only to be later reconquered approximately sixty years later, shortly before the conquests of Alexander the Great).

It should not be surprising that there were many similarities between the Assyrian and the Persian Empires. In many ways, the Persians were inheritors of the Assyrian legacy. As such, they continued many of the practices and functions of the previous empire, although they did institute their own as well.

Both empires were known for their strong and efficient militaries. Territorial conquest was important in both empires and their militaries incorporated soldiers from all parts of their territories. More specifically, both empires made use of archers who were well trained and made up a large cohort of any fighting force. This strong military was essential. There were few natural barriers in their territories that they could rely on for defense and being located in the fertile crescent and the junction of three continents made their territory covetable by others.

Since they occupied much of the same land, both the Persians and the Assyrians practiced similar agricultural techniques. They employed extensive and complex irrigation systems which made use of the water from the Tigris and Euphrates Rivers. They both cultivated large amounts of grains which would be stored in times of plenty in anticipation of years of famine. As such, they were able to provide a reliable food source to a large population.

Both the Persians and the Assyrians were avid record keepers. Archaeologists have found countless accounts written in clay tablets from both empires recording everything from mundane business transactions to the legendary exploits of their kings and warriors.

The Persians and Assyrians were both large empires in the Middle East from the Iron Age world. Both kingdoms boasted powerful militaries that protected their vast holdings and expanded their borders. The empires were not protected by natural barriers so a strong military was a necessity. Despite this geographical shortcoming, the empires utilized arable land, both being located in the fertile crescent around two rivers.

The two empires were both ruled by monarchs. The empires employed organized bureaucratic systems to enforce laws and collect taxes. Both empires are mentioned in the Old Testament, or Torah as having conquered the Hebrews, although the Persians are painted in a more positive vein in the scriptures. While both empires were known for their decisive military strength, both treated their new subjects with tolerance as long as the new citizens paid their taxes.

What are the differences between the Persian and Assyrian empires?

The Assyrian Empire enjoyed a much longer history than the Persian. It was one of a collection of nation-states in the ancient Near East that persisted more or less continuously from the 3rd millennium BC until its spectacular fall in 612 BC. Ancient Assyria was located in northern Mesopotamia, bordered to the south by Babylonia, to the north by Mittani and the Hittites, and to the west by Egypt. These states traded and warred with one another during much of the 3rd and 2nd Millennium BC.

With the rise of the Middle Assyrian Empire in approximately 1400 BC, however, the balance of power began to shift and Assyria began to dominate its neighbors. Assyria conquered Mittani and Babylonia during this period, and only an alliance between the Hittites and the Egyptians prevented their further spread. During the Bronze Age Collapse, Assyria survived the incursion of foreign invaders that marked this period and entered the classical period of antiquity in a strong position. During the Neo-Assyrian Period beginning approximately in 900 BC, the Empire once again spread over the entirety of Mesopotamia, but this time went further, conquering the Levant, much of Anatolia and Persian territories to the east. Incredibly, during the last century of its existence, the Assyrians even conquered Egypt and their empire reached its greatest extant.

The Assyrians were trailblazers. They developed the first professional army in history, which was heavily reliant upon the chariot. With this large, well-trained army, the Assyrians conquered the first multinational empire in history. They maintained control over their subjects through terror. Assyrian reliefs taken from palaces in Nineveh and Assur relate gruesome tales of mass executions and describe the destruction of cities like Babylon almost gleefully. These acts not only helped quell dissent but also furthered Assyria's expansion as enemies who resisted could expect brutal subjugation, as was the case of Babylonia, or outright extermination, as in the case of the Elamites and the Mittani. So many opened their gates when the Assyrians came in force.

The downside of rule through terror became apparent with the sudden fall of Assyria in 612 BC. Assyria's ruin was actually brought on by its expansion; the empire became the first victim of imperial overstretch, possessing more territory than it could actually administer. Further, expanding its borders brought Assyria into contact with new enemies such as the Medes, Cimmerians, and Persians, who were strong and practiced novel ways of fighting. A coalition these enemies was joined by a resurgent and vengeful Babylon in 612 BC, and in quick succession, the great cities of Assyria were sacked violently and the Assyrian Empire was destroyed, never to rise again as a player in the Near East.

Assyria laid the groundwork for the Persian Empire by unifying and subjugating much of the land that the Persians would later conquer. In a sense, Assyria softened up much of the region to the conquests of the Persians, Greeks, Romans, and Arabs. Cyrus the Great, the founder of the Achaemenid Dynasty, led a revolt against Medea, one of the Kingdoms that had conquered Assyria only a half century earlier. He would then, within one lifetime, conquer much of the territory it had taken Assyria centuries to acquire. His descendants would reconquer Egypt and extend Persian control into central Asia and India, creating the largest land empire in history up to that point.

As rulers, the Persians generally had a lighter touch than the Assyrians. The Persians named regional governors over their conquered territories and allowed conquered lands to maintain a certain degree of independence. The Persians were certainly capable of using terror as a ruling strategy, but it was not their preference, and they didn't revel in it the way the Assyrians did. The Persians also did more to foster commerce and cultural interaction through the building of sophisticated road networks and the development of coinage.

In contrast to Assyria, when Persia fell, it was not an internal uprising or a coalition of enemies that brought it down. Rather, Alexander the Great led one of the most remarkable military campaigns in history, essentially conquering the Persian Empire whole after a series of brilliant victories. That Alexander kept much of the Persian infrastructure and bureaucracy in place is a tribute to how well it functioned. That, rather than supplanting Persian culture with Greek, Alexander attempted to fuse the two shows that he admired much about his newly conquered subjects. We see no such admiration for the Assyrians. One of the most important sources regarding the Assyrian Empire is the Hebrew Bible, which essentially curses the Assyrians for Sennacherib's conquests of the Kingdoms of Israel. When Assyria fell in 612 BC, the Bible's condemnation states "Who shall bemoan her?" Cyrus the Great, on the other hand, is praised in the Bible for ending the Babylonian Captivity and allowing the Jews to return to Israel.

The difference between the Assyrians and Persians can best be understood in the context of their treatment of the Hebrews. The Assyrians conquered the northern kingdom of the Hebrews, Israel, in the Eight Century BC. They brutally massacred all of their leaders and captured the rest of the population and brought them back to Assyria as slaves. The Assyrian action was done to squelch any hint of rebellion within the empire. The Assyrians developed a reputation for brutality and torture of its enemies. The government was centralized with a powerful king that believed everyone under him was at his disposal.

The Persians, on the other hand, were more progressive in their approach towards their subjects. When Cyrus (560 BC) acquired the Babylonian Empire, he treated his subjects with a certain degree of autonomy. The Hebrews were allowed to return to their homeland and rebuild Jerusalem and the temple. A high priest was appointed as well as governor from Persia to rule the land. The Persians divided their empire into provinces, called satrapies, which gave local rulers a greater degree of flexibility. Cyrus did not necessarily allow the Hebrews to return out of the goodness of his heart. He felt that the Hebrews would work hard to restore Palestine and make it profitable for his empire.

When looking at the Hebrew example, it is clear that the Assyrians and Persians believed in different methods of control over their conquered subjects. This was a major difference between two kingdoms that otherwise had similar goals.