Suppose I have the random variables $X_1, .., X_n$ with the marginal distributions are not normal (in fact, unknown marginal distribution).

Will there be any differerence between the assumption $X_1, .., X_n$ are independent and $X_1, .. X_n$ are modeled with t-Student copula with the correlation matrix is all zeros (with 1 in the diagonal) ?

Thank you for your help!