Have a son each at Yale and Stanford, and I don’t really buy the whole ‘innovative vibe’ stereotype, and definitely wouldn’t make a decision based on that. Both schools have pretty enthusiastic, quirky cultures, though most students are actually highly driven underneath. For every new startup at Stanford there is a creative new theater production at Yale and vice versa.
Both schools have have very strong Poli Sci Departments, so academic strength is also not a distinguishing factor. I daresay that a greater mass of students at Yale tend to be politically/philosophically engaged which means that there is a greater mass of students who share the same interests as you. For instance, groups like the Yale Daily News, Yale Political Union, or the Buckley Program are highly developed and active.