New Haven is crap, but Yale has the secret societies and the weight of British-style academic tradition. It's by far the quirkiest of the three.
Princeton is smaller and in a cuter town, it has more STEM, athletics and parties than Yale.
Harvard is Harvard. It's the most famous institution on earth. Also the most cut-throat in both undergrad and grad schools. I am a foreigner who has lived in Europe and Asia. Most people haven't heard of ***ANY*** US university apart from Harvard. I know this will be hard to hear for some of you