"Prosodic phonology" is ambiguous, since it is used to refer to a specific generative theory of structural relations, as well as numerous often non-generative accounts of non-segmental phonology (as discussed here). Setting aside the non-specific version of "prosody", "prosodic phonology" refers to the hypothesis that there is a strict hierarchy of structural constituents above the segment. They can be divided into two subsets, as exemplified here, with certain additions (mora at the bottom, possibly but probably not sub-syllabic constituents like "onset", "rhyme" etc. in the middle, also possibly "Clitic group" immediately above the Prosodic Word. Prosodic Phonology (PP) is a specific claim, that these structures exist, they are ordered in a specific way, and the hierarchy is strict (IP always and only immediately dominates PhoP, etc): though, some people give up on that requirement. The primary purpose of PP is to provide a mean of handling syntactic-grouping influences on phonological rules, for which purpose PhWd and higher were invented. Structure below the word exists for entirely different reasons.
Metrical phonology comes in two main flavors. In the earliest manifestations e.g. "Fragment of a draft", it is a formalism that competes with autosegmental theory, as a means of handling all of phonology. Autosegmental phonology did not attempt to handle stress, and metrical phonology was initially born to account for stress and rhythm. The formalism metrical phonology was later generalized so that it could handle feature spreading (via the notion of "projection"). Leben's paper "Metrical or autosegmental" pointed out that there wasn't an empirical issue, and the extensions of metrical phonology that allowed MetPho to handle all of phonology were withdrawn. At that point, MetPho became a more restricted theory of grouping, with syllables, feet and P-words (plus occasionally-postulated binary groups of feet). The theory was mostly "about" stress, and has attempted to express the notion "is more prominent" in some way. There is considerable overlap between the lower half of the ProPho tree and MetPho" syllables group into feet which can group into a word-size chunk (so that one can say that the word has main stress on a particular foot).
Autosegmental phonology on the other hand is a theory of features and their relationship to segments. Thus it has minimal overlap with metrical phonology. However: autosegmental phonology can refer to metrical or higher structures, if e.g. a tone spreading rule is sensitive to foot structure or syntactic structure (translated into PhoPhr structure). The three theories are not competitors, they are about different things (just as botany, zoology and geology are about different things, and it makes no sense to wonder if botany is more explanatory than geology). Prosodic Phonology is really about one specific claim, that phonological rules cannot refer to non-phonological properties, and the phrasal machinery (and more) is what was thought necessary to handle the facts. Some version of Metrical Phonology is how people handle stress and foot structure, and some version of Autosegmental Phonology (and its more specific descendant Feature Geometry) is how people handle things that aren't stress and foot structure. Metrical phonology is about grouping sequences of segments into higher units, and Autosegmental Phonology is about what you do with features.
The "not isomorphic" claim is really a red herring promulgated by proponents of translation theories of syntax-phonology relations. The claim is that substrings of sentences where phonological rules apply are not syntactic constituents. This is a red herring because people who advocate the Direct Reference theory never claimed that rules only apply to the entirety of a syntactic constituent: the claim is that rules refer to syntactic relations, such as c-command.
