Different assumptions between independent normal and multivariate normal distributions

I have been dealing with multivariate meta analysis lately. In the literature the various authors say that the main assumption of the MVMA is that the different variables are Multivariately distributed. I understand that. However I don't understand how that assumption is stronger than the assumption that the different variables are independently normally distributed. Could somebody explain that?

• Marginal normal independence is simply a special case of multivariate normal with zero covariance. L – SmallChess Feb 21 '17 at 0:26

Basically, if you assume that your data is distributed as independent normals, then you also assume that the variables are uncorrelated. On another hand, if you assume that they are distributed as multivariate normal, then it also means that you consider that they may be some non-zero correlation between the variables (recall that multivariate normal is parametrized by covariance matrix $\Sigma$).