We present a comparison between a global energy approach à la Griffith and a gradient-regularized damage formulation named graded damage for simulating quasi-static fracture in brittle and quasi-brittle solid materials. Though intrinsically different, the two methods share a number of common features emanating from the underlying potential structure. In the Griffith-like approach the fractured state of a solid is obtained from incremental stationarity of the total free energy resulting from the sum of the bulk elastic energy and the surface energy required for crack advancement. On the other side, the graded damage model inherits the variational properties of generalized standard models, whereby the solution of the evolution problem amounts to minimize incrementally a global energy potential with convex inequality constraints. A numerical example that refers to a typical benchmark problem is presented that allows showing the features of the two approaches and their ability to reproduce experimental results as well.