When using domain decomposition in a finite element framework for the approximation of second order elliptic or parabolic type problems, it has become appealing to tune the mesh of each subdomain to the local behaviour of the solution. The resulting discretization being then nonconforming, different approaches have been advocated to match the admissible discrete functions. We recall here the basics of two of them, the Mortar Element method and the Finite Element Tearing and Interconnecting (FETI) method, and aim at comparing them. The conclusion, both from the theoretical and numerical point of view, is in favor of the mortar element method.