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).
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3$\begingroup$ I agree in general that too close tags should be made synonyms (another example recently mentioned being lower-bounds and inequalities, see meta.mathoverflow.net/a/4907/14094) $\endgroup$– YCorCommented Mar 21, 2021 at 19:24
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$\begingroup$ @gvmh: The problem is that the Wiener integral is a "path integral", too; should we make it, too, a synonym of "path-integral"? Furthermore, unlike the Wiener integral, the Feynman integral is not a "true" integral. Organizing MO is important, but we should strive not to be zealous in doing so. I would keep these tags as they are now. $\endgroup$– Alex M.Commented Mar 25, 2021 at 0:02
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$\begingroup$ @AlexM.: but then the usage info and tag wiki for path-integral needs to be updated to reflect that greater generality. Currently, it (somewhat vaguely) describes the Feynman path integral. $\endgroup$– gmvhCommented Mar 25, 2021 at 5:44
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$\begingroup$ @gmvh: Since there are currently 17 questions tagged with "feynman-integral" and only 5 tagged with "path-integral" (of which one is about the Wiener measure!), I would make "path-integral" a synonym of "feynman-integral" and I would merge the tag information and tag wiki of "path-integral" into the tag information and tag wiki of "feynman-integral" (with the necessary modifications, of course). $\endgroup$– Alex M.Commented Mar 25, 2021 at 8:12
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1$\begingroup$ @gmvh: But! There is a "but"! Feynman integrals are "integrals" over spaces of paths only in quantum mechanics. In quantum field theory, they are "integrals" over spaces of fields, so in this case the connection with the "paths" breaks down. This is why I suggest to leave things as they are (and I shall maybe post this as an answer). $\endgroup$– Alex M.Commented Mar 25, 2021 at 8:14
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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.
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$\begingroup$ But fields on $\mathbb{R}^{n+1}$ are also paths, viz. in the space of fields on $\mathbb{R}^n$. $\endgroup$– gmvhCommented Mar 28, 2021 at 14:29
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1$\begingroup$ First, nobody that I know in this area thinks about fields in the way suggested by you. Second, quantum field theory can be studied on space-times that are not necessarily products of the form "space $\times$ time". $\endgroup$– Alex M.Commented Mar 28, 2021 at 15:46
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$\begingroup$ Well, pedagogical treatments of lattice field theory quite often introduce things that way (it links quite naturally with the transfer matrix formalism). But it is of course perfectly true that the functional integral formalism also works on manifolds where no such split is possible. $\endgroup$– gmvhCommented Mar 28, 2021 at 18:07