MSAA: Costs memory space and memory bandwidth. If you are going to antialias a pixel with, say, 6 sample points (6X AA), you will need to reserve memory for all 6 sample points, and there will be at least some pixels where you have to access all 6 sample points during rendering. Also, with MSAA, if a pixel lies on the edge between 2 polygons, it needs to be rasterized twice, once per polygon, whereal without MSAA you would pick 1 polygon and thus rasterize the pixel only once, so MSAA costs a bit of pixel fillrate as well.
AF: Cost depends on method used. For the Feline method or approximations thereof (which AFAIK are the most common methods used nowadays), you will be sampling the texture a variable amount of times per pixel and adjust the mipmap selection so that you sample from a more detailed mipmap - costing you a great deal of texture memory bandwidth and quite a bit of texturing fillrate. There are other methods, but with other costs; e.g. Rip-mapping is cheap in terms of fillrate/bandwidth even for high degrees of anisotropy, but requires 3 times as much texture memory as the Feline method and suffers from severe angle-dependency.