An important point that has not been addressed in the previous (excellent) answers is the actual estimation step. Multinomial logit models can be estimated numerically because their CDF has an analytical integral. The density function of the normal distribution is not, so probit models require simulation. So while both models are abstractions of real world situations, logit is usually simpler to use on larger problems.
That said, there has been some interesting work by Chandra Bhat in finding fast estimators for general probit models, if you're interested.