A multivariate normal distribution can be thought of as normal margins with a normal copula. The multivariate t is the same way, but it has t margins with a t copula and they all have the same degrees of freedom. So it has t copula dependence. It is either a spherical or an elliptical distribution.
I can't think of a good reason to use a multivariate t. The degrees of freedom parameter is the same for both univariate and multivariate. You give yourself more options by fitting univariate t distributions for each series (so they could potentially have different degrees of freedom, more important when you're looking at a variety of different assets or factors).
The dependence can be handled by a copula. The normal copula is easy to use, but if you're concerned about tail dependence, then there are a number of options, including the t copula. Again, as with the multivariate t, the t copula has one degree of freedom parameter, which means that the tail dependence is assumed to be the same for everything. So if you're looking at data where it makes sense for all to be the same, perhaps such as U.S. equity sectors, then this is fine, but in the majority of cases it may not.
As an alternative, some people use grouped t copulas (where every series gets its own degree of freedom) and others use vine copulas (which allow an even greater potential for combinations to explain the dependence structure). One downside is that the more complicated the copula, then the more time-consuming it is to fit, especially if you want to fit a DCC copula. This also illustrates the importance of dimension reduction. I try to avoid just putting everything in a big multivariate distribution.
If I'm setting up a quantitative portfolio management model, more complicated tail dependence is pretty low on my list. I recall some benefit in backtests, but it was tiny compared to incorporating time-varying variance. You might be looking at different data, so do your own research and evaluate.