Mosek is an actively developed commercial solver. SeDuMi and SDPT3? are non-commercial solvers whose development other than minor bug and compatibility fixes ended years or decades ago. In fact, SeDuMi’s developer died over 2 decades ago, although CVX developer @mcg and others did make some bug and compatibility fixes incorporated in the SeDuMi version included with CVX. I will defer to Mosek employees @Erling , @Michal_Adamaszek o @hfriberg to discuss further.
As for TFOCS vs. Mosek, I have no idea, but feel free to share your experience if you get TFCOS implementation in good shape (or maybe you don’t see the need now that you have used Mosek?). I don’t think TFCOS has been further developed for the better part of a decade. As you can see, TFOCS developer @Stephen_Becker just posted in your other thread TFOCS for mixed trace and $\ell_1$-norm minimization .