What is the difference between political science and politics?
The term "political science" derives from the Greek word "polis" (plural: "poleis") which means "city state". In Greek philosophy, human affairs and the science of understanding them are divided into three categories, ethics, which studies the human as an individual, economics, which studies the individual household (from "oikos" or household, only later taking on the sense of finance), and politics, which approaches humans as they interact in larger social groups.
The main distinction between the terms "politics" and "political science" is that the former refers to the actual process of how humans interact in groups and the latter the scientific study of the former. Aristotle, who is the origin of much of our terminology in this matter, defines humans as "political animals" meaning that we tend to form large social groupings but refers to the science of studying these groupings as "politikê epistêmê" (political science).
Politics constitute an activity, e.g. "John's response to Mary's report was based as much on office politics as on close analysis", but political science is an abstract system of knowledge, e.g. "Political scientists are increasingly using 'big data' to correlate voting and shopping patterns."
Political Science, as one other answer notes, is an academic discipline. However, it is far broader than simply the study of politics. Political Science does involve the study of politics, but it encompasses also the study of political systems around the world, including the structure and functioning of disparate governments in every region of the world. Governing structures vary widely around the world, especially during the Cold War, when the Soviet Union and its satellites and allies in Eastern Europe and across the so-called "Third World" were governed by brutal totalitarian regimes that often had forms of parliaments that were simply "rubber stamp" institutions that provided a thin veneer of respectability for the dictators who actually ran those countries. In other words, communist regimes had "central committees" and other ostensibly legislative bodies that, in practice, merely affirmed decisions made by the top handful of members of those governments' ruling circles, usually known as a "Politburo," or Political Bureau.
Within the narrow definition of "Political Science" provided in the other answer, the field does, as noted, involve the study of politics, such as the factors that cause people to vote a certain way in elections, public confidence in the governing structures, and divisions within electorates that help to understand the perceptions of different ethnic groups or socioeconomic classes toward various issues and politicians.
"Politics" refers to the practices of elective and nonelective political systems with regard to the manners in which leaders are elected or appointed, the minutiae involved in political campaigning in democratic systems, the dynamics between and within disparate factions within any particular system (for example, relations between so-called "hard-liners" and "liberals" or "moderates"), and the relationship of citizenry to governments. In a democratic system, politicians (those seeking elective office such as mayor or congressman) interact with the public whose votes they need to win an election. In a nondemocratic system, the focus is more on relationships within the governing party or regime. All of this constitutes "politics." Political Science is the study of all of this, as well as of the economic systems of different countries and the manner in which relationships with other countries or political entities are conducted.
The main difference between these two things is that politics is something that happens in the real world and political science is the study of that thing. In other words, political science is the study of politics.
When we are talking about government (as opposed to something like politics within a workplace), politics is the process by which people try to influence their government and the process by which the government decides which policies will be enacted. These are activities that go on every day at multiple levels of government.
By contrast, political science is an academic discipline. It is the study of politics, not the actual political process.