Wireless channels may be constantly and exponentially correlated (I'm interested about antenna correlation only). Actually what is the difference between these two correlations?
I think you mean by constant model the uncorrelated model. This model is very simple and can be modelled using the rayleigh fading channel. With this model the eigenvalue are constant and all equal. In the exponential correlation model, there is a correlation in matrix and the eigenvalues are not equal. However, since the degrees of correlation decay exponentially it called exponential correlation model.
Of course, if you want more practical channels, then you need to use channel correlation. The are many different channel correlation models. However, exponential correlation model is used quite often in the literature A. S. M. Badrudduza .