Yes there is a difference. In multivariate probit there are more than one binary outcomes and hence multivariate.

But in multinomial probit model there is only one outcome which is not binary but has more than two categories. That is the difference. I attached corresponding Wikipedia pages.

https://en.wikipedia.org/wiki/Multinomial_probit

https://en.wikipedia.org/wiki/Multivariate_probit_model

Like Mehmet says above: multinomial means the dependent variable (outcome) has more than 2 levels, multivariate means there is more than one dependent variable (outcome).

thank you so much