I think the differnce in location is very much to your personal preference, with regards to which area you will like.
The OP is correct about the 20 minute train ride, but if we look at the areas as exclusive from one another (which they're obviously not), I think you can easily say that NYU offers an atmosphere that Columbia does not, and that Columbia does the same.
Personally, I don't think there is a substitute for experience, and I'm not sure how you can gain an understanding for Morningside Heights if you don't live it. You are very close to a culture mix, and while all of Manhattan is a huge culture mix, you don't often feel a culture clash so much until you get up around 125th street on the Upper West Side.
If you're not into that, and you're looking for fun, expensive bars, in the midst of the capital of "artsy", then NYU is probably a better fit, but again, no one, especially myself, is an expert, so maybe an extended visit to Manhattan isn't the worst idea.
It's a moot point for me, but just my two cents.