In a sense, approaches to international relations are grounded in assumptions about human nature writ large. These two different theories are grounded in quite different assumptions about people in their "natural state." It should be noted that the use of the term "liberal" in international relations refers to a sense more common in Europe than the United States, deriving from its Latin root "liber" (free), meaning a position supporting free markets and limited government regulation.
Realism begins in the work of Thomas Hobbes, who in his Leviathan famously claimed that the state of nature was "bellum omnium contra omnes" (war of all against all). Hobbes assumes that people act entirely out of self-interest and are always competing for resources. States exist as people band together for safety and security and face off against other states. Strong, authoritarian states are needed to rein in people's naturally bad tendencies. Rather than believing that people can work together to attain mutually beneficial goals, "realists" think multilateralism is at best a screen for self-interest and that international relations are a competition in which there are winners and losers. Thus, while realists may be successful in attaining narrow objectives, they tend to be unrealistic in their assumptions about human nature and ill-equipped for dealing with issues such as global climate change, public health, or migration where cooperation and global thinking are needed.
Liberals tend to believe that human nature is not uniformly bad but rather that people have benevolent as well as selfish impulses. While we may try to get ahead at work, we might donate time or money to charity, help our friends and family, or even send food or money to help victims of natural disasters in other countries. Rather than seeing authoritarian states as good, liberals believe that free markets and open societies have better outcomes. The liberal democracies of the west, liberals note, are among the world's most prosperous nations. They see international relations not as a struggle with winners and losers but rather as a way to work multilaterally towards a mutual prosperity. Liberals are focused on economic development, seeing increasing prosperity as removing many of the causes of conflict.
On migration, for example, realists believe that building walls or enforcing borders with military force is a solution. Liberals argue that this has proven ineffective, expensive, and inhumane. Instead, one should focus on international development. If people can live good lives in their own countries, free of oppressive regimes with decent job opportunities, fewer will try to migrate.
While realism and liberalism have some similarities, their differences are much more important.
Both realism and liberalism believe that the world is a dangerous place. Both realize that countries can go to war with one another and destroy one another. Both realize that there is no world government in place that can prevent countries from doing harm to one another.
Both realism and liberalism believe that military power is important. Both sides understand that states can use military power to get what they want. They are both aware that countries without military power can be abused by other countries.
While these are important similarities, they are less important than the differences. First, realists believe that military power is the only relevant kind of power. Realists do not believe that things like “soft power” can be at all important. By contrast, liberals believe that military power is only one kind of power and that it is not even the most important kind of power in many cases. Instead, they believe that things like economic power and moral power can be more important than military might.
Second, realists believe that there is no point to international institutions because they cannot force countries to comply. They believe, for example, that the United Nations is pointless because it cannot keep a country from doing what it wants (say, for example, that it cannot prevent Russia from annexing the Crimea). Liberals understand that the UN cannot force countries to obey, but they believe that it is still very important. They believe that international organizations give countries ways in which to cooperate with one another and to gain one another’s trust.
The last major difference between the two that I will address here is the two theories’ different views on what states want. Realists argue that all states have the same interest. All states are interested in increasing their power in the world. Liberals do not agree. They say that there are many states that really do not try to increase their power (remember that power is only defined for realists in military terms) and that we cannot understand other states unless we understand what it is that they want.
These differences (along with many others) greatly outweigh the similarities between these two theories of international relations.