I'm watching philosophy crash course, and the narrator claimed that Utilitarianism is the antithesis of the of Kant's categorical imperative. However, on the surface, I can't see a major difference. How do the two differ in their approaches?
Long post, sorry! As a person who has made a few LD debate cases framed around both philosophies, I have a lot to say and will do my best to answer this. However, since I'm not a philosophy student, you may take my words with a grain of salt if a more knowledgeable person corrects me (and I highly encourage anyone who wants to correct me to do so). Also, I'll assume you're referring to act utilitarianism (the standard version) for the purpose of this post.
As you probably know, act utilitarianism is based on the idea that given a decision, one should only take the action that results in the most amount of "utility". It's basically about maximizing the amount of happiness in the world / minimizing the amount of suffering, and how you get there is irrelevant. On the other hand, Immanuel Kant's philosophy is centered around the idea of the "categorical imperative." According to him, there are 3 formulations of the categorical imperative, and Kantianism is based on obeying those 3 formulations of the CI, regardless of the consequences. I will go through the first and second formulations and how those clash with utilitarianism, and leave out the rest of Kant's philosophy because it's simply too much to put in one post.
The first formulation is that one should "act only in accordance with that maxim through which you can at the same time will that it become a universal law." Basically, what this means is that you should only do an action if you would want every person in existence to follow that action as moral law. This is essentially a way of determining whether the action is moral, regardless of the consequences. For example, if you could kill one person to save five people, you should not do it because if everyone were to freely kill one another society would fall apart (and you don't want that). Utilitarianism, on the other hand, focuses only on the consequences and evaluates each action individually. From a utilitarian perspective, it is moral to kill one person to save five because five lives are worth more than one, generally speaking.
The theories also differ when you begin to look at Kant's second categorical imperative, which states that one should treat humanity always as an end and never as a means only. Basically, it means that one should never consider any other factors besides a person when determining how to treat that person. If you could falsely imprison someone to stop violent rioting in a town, you should not do so because you would be using that someone for another purpose (treating them as a means for the goal of peace). Rather, you should treat them as an end, and determine whether they, as a person, deserve to be imprisoned. Since that person is innocent, they do not deserve punishment, and to treat them as ends means you cannot imprison them. Obviously, utilitarianism takes the opposite view, as mass suffering could be prevented by falsely imprisoning only one person (and utilitarianism is based on minimizing suffering / maximizing happiness).
Both philosophies are a lot more complicated and have much more conflicts, but that requires a lot more in-depth analysis of both theories and hopefully the point is made by now. Kantianism focuses on the action and utilitarianism focuses on the consequences. At the very core of this is the debate between consequentialism and deontology, and of whether the ends do or do not justify the means. Hope this answers your question without too much confusion.
Resources on both philosophies if you want to read further: https://www.iep.utm.edu/util-a-r/ https://plato.stanford.edu/entries/kant-moral/#CatHypImp