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I've Googled several sources that compare two of these, but none that contrast all four:
Political Philosophy with Political Theory.Political Science with Political Philosophy.Political Theory and Political Thought. I haven't asked about the latter, as it looks to be a rarer term.
As a bit of disclosure, I am a political scientist and have done work in political theory. By way of establishing my credentials, you can find my M.A. thesis online here. The claims below will be generally unsourced, because they are part of the professional knowledge of a political scientist.
This is actually the hardest to define. The most common definition is something like, "the things that we all have in common". The idea is that the actions of the government should be important to all of us, whereas the actions of private individuals and businesses are really not the public's business.
More practically, politics is whatever the government and politicians do. It involves how we choose leaders (elections, wars of succession, etc.), what leaders do (legislate, interpret laws, enact deals with other leaders), the formation of political units (nations, cities, parties), the activities of political actors (voters, juries, government officials, etc.)
Political science is simply a scientific discipline which investigates politics. The key here is 'science'. Political scientists are (typically) not interested in moral or ethical claims. Our goal is (typically) to advance our understanding of how politics works.
How does this happen? We formulate theories that attempt to explain how certain things work. These theories make testable predictions (called hypotheses). We test these hypotheses using a variety of methods including case studies, interviews, surveys, statistical analyses of data, etc. Based on these tests we refine our theories. Over time, we end up with a more robust understanding of the subject.
Somewhat confusingly, not all "political scientists" are "scientists". A major trend right now is work in critical theory, which comes to us from the humanities. Critical theorists do not offer testable hypotheses, but attempt to provide descriptive insights in a more philosophic way. Theory (including critical theory) is useful to scientists for helping us explore theories and generate novel hypotheses.
Political philosophy is any kind of philosophy which deals with politics. Examples include classics like Aristotel's "Politics", Hobbes' "Leviathan", and Adam Smith's "Wealth of Nations" as well as innumerable others.
Political philosophy is not a science; it does not offer testable hypotheses and it cannot be tested against the world. Like political science, political philosophy is a way of explaining the how the political world works (that is, both of them are a different approach to studying politics).
Many political science departments will have political philosophy specialists. This is partially for historical reasons, but also because their subject matters are very similar, even though their approaches are different.
Political theory encompasses two different things.
First, it is often a synonym for political philosophy. Easy enough.
Second, political theory can refer to the theories generated by scientific research. For example, one theory of voting behavior is the Michigan Model. If we talk about the theory without creating any testable hypotheses and evaluating them, we are engaging in political theory.
I don't think this is a definitive explanation, but might serve you well.
Political Philosophy begins first with the Philosophy of Ethics: i.e. what is the "good". Political philosophy is how to secure or achieve that good once identified through the organization of man's endeavors, i.e. politics. Many Political Philosophers were first Ethical Philosophers or have ethics (or a type of ethics to be exact) baked into their political thought: Plato, Aristotle, Hobbes, Locke, Spinoza and Hume, to name just a few.
Political Theory is taking the abstract of political philosophy, and applying it to a specific circumstance that includes and is influenced by a cultural, religious and historical heritage. For example American Political Theory had its beginning in the Enlightenment but was influenced by the Puritans, as well as by the writings of Frederick Douglas. So the idea of America of being the "City on the Hill" can be traced back to all of these antecedents.
Politics can refer, as stated above, just to how men organize public affairs and governance. Or it can refer specifically to the pursuit of power among politicians and interest groups.
Political Science is in fact a loaded term and is contrasted with Government. Both are the study of politics. However, political science invokes a few concepts that conservatives, neo-classicists, and traditional liberals (in the sense of the Enlightenment, sometimes called neo-liberals) would seriously disagree with, and it's how they view man.
Progressives, and those that favor the term political science, will view man (and their institutions) as improvable, not from an evolutionary standpoint, well not exclusively, but from the explicit call for historic moral improvement in a Marxian sense. With better and more rational programs and government, man will eventually achieve Utopia. The 60's programs often called for a host of statisticians, nutritionists, psychologists and others to figure out how to just "improve" man "scientifically". Most of the 60's programs, like the War on Poverty, had very mixed results, with result that such Progressives were in some ways overtaken by the "New Left".
See for example this Wiki explanation for the rise of social and political sciences
Many activists joined efforts to reform local government, public education, medicine, finance, insurance, industry, railroads, churches, and many other areas. Progressives transformed, professionalized and made "scientific" the social sciences, especially history,[4] economics,[5] and political science.[6] In academic fields the day of the amateur author gave way to the research professor who published in the new scholarly journals and presses. The national political leaders included Theodore Roosevelt, Robert M. La Follette, Sr., and Charles Evans Hughes on the Republican side, and William Jennings Bryan, Woodrow Wilson and Al Smith on the Democratic side.
Or this longer expose from the Heritage Foundation on Charles Merriam explaining progressive political science
Charles Merriam, a longtime professor of political science at the University of Chicago and political activist in the first half of the 20th century, developed a new “scientific” study of politics that served the Progressive cause.
Government advocates view man's nature as much more immutable based upon mostly Christian canon, that man is naturally defined as lower than the angels but higher than the beasts. (One reason the Christian god of the Enlightenment is sometimes referred to as Nature's God). They didn't see any real path for improvement, and were more concerned therefore about the rise of despotism from government sources. Hence separation of powers and limited government and other concepts you see from Enlightenment thinking. There is no science here. The study of government is reductive to philosophy, rationality, realism and better understanding of man's nature, what he's capable of, and the benefits and drawbacks of culture and tradition. Government is therefore a study in art. In the same way that the New Left threatens Progressive thought, the popular strains of Trumpism so threaten conservative thought.
See for example James Madison's famous Federalist locution:
If men were angels, no government would be necessary. If angels were to govern men, neither external nor internal controls on government would be necessary. In framing a government which is to be administered by men over men, the great difficulty lies in this: you must first enable the government to control the governed; and in the next place oblige it to control itself.
Note this is one reason some colleges have political science departments and others have government departments.
Further note that Political Science can also just refer to using metrics to measure polling and other applied technical empiricism, not to confuse you more.
