The question is in the title: I see no benefit from having these two tags for essentially the same concept (particularly as the path-integral wiki excerpt refers to the path integral formulation of quantum mechanics, rather than to any more general notion of path integration).
No. While it is true that in quantum mechanics a "Feynman integral" is a "path integral" (i.e. an integral of functions defined on a space of paths), in quantum field theory Feynman integration is performed on functions defined on a space of fields - so no paths in this case.
In fact, you could try the opposite: make "path-integrals" a synonym of "feynman-integrals" (since the former are a particular case of the latter). This wouldn't be fine for a different reason, though: Wiener integration is yet another integration on spaces of paths, but it is not Feynman integration. (Furthermore, while Wiener integration is rigorously defined (a true Lebesgue integral), Feynman integration is not, which complicates matters.)
I suggest leaving these tags as they are.