What are the similarities and differences of the Hellenic and Hellenistic periods of Greek history?
I think by "Hellenic history", the questioner probably means Greek history before the so-called Hellenistic Period, the latter of which dates from 323 BCE (the death of Alexander the Greek) to 31 BCE (the Battle of Actium).
In both periods, the Greeks must deal with the issue of the Persians. In the 490s and 480s, the Greeks were trying to repel the Persian invasions, and one of the major reasons given for Alexander's campaigns in the 330s was to take vengeance upon the Persians for their incursions into Greece in the previous century, as well as the continued Persian control of Ionian Greeks living along the coast of what is modern Turkey.
Prior to the rise of Macedonians in the 300s, the Greeks did not have anything resembling a united Greek government. They lived in city states, each of which had its own form of government and was an independent political entity. Alexander's father Philip helped bring about the fall of the city state. After his defeat of the Greeks at the Battle of Chaeronea in 338 BCE, the Greeks came under the control of Philip, then Alexander, and then Antigonus and his successors, until the advent of the Romans in the 100s BCE.
From an artistic standpoint, we might say that before the rise of Alexander the Greeks appear more idealistic, whereas during and after the time of Alexander the Greeks seem more realistic. A comparison of the famous Doryphorus statue with the Dying Gaul provides one example of this.
Well there are three I know off the top of my head. First, the hellenstic kingdoms was divided into almost nine geographical areas, yet it spands from Macedonia all the way to egypt, yet the kingdoms where not organized as even Alexander the great wasn't a great politician, yet he was more of a military leader, so as you go further when he conquered the persians he influenced hellenism, yet he did intertwine both the persian god's with the greek hellenistic gods. The first hellinestic period was actually really short started after the death of Socrates where Hellenistic kingdoms began to flourish across the meditteranean thus the influence grew stronger. The first period was in fact very primitive as only they had such little land. Then the Death of Plato in 347 soon Plato would create the Hellenistic Religion and spread the words of the gods across the meditteranean. Then the birth of Alexander the great in 356 B.C.E soon Aristotle dies in 322 the successor of Plato some say. Then Alexander conquers egypt and claims soul authority in 3rd century B.C most people say. Then at the height of his power the Maccabean Revolt in Judea 127 B.C it was not successful. Soon the Roman empire is at the height of its power and takes over the helenistic kingdoms. As much as I can say, the difference between the kingdoms is actually they where unorganized which Ceaser Augustus in fact created the actual first Democracy destroying the Senate (which ran the military) creating a military government. Most of the helenistic kingdoms where actually a military government, yet some yet few where governed by the people. Also much of the Hellenistic kingdoms did was help obtain power by conquering villages forcing people to join. though many people thought the idea of Mixing religions was in fact immoral. Actually soon the rise of Monotheism would end that entire system of government. The Hellistic kingdoms did have emperors or pharohs. It was a diverse empire really. There are even greek statues of Bhuddah found all over Aisia. Even cultures that morally hated each other used each other's architectures.