I understand that it is a prosodic word that dominates feet in a classic hierarchy -- but why do we need to refer to a prosodic word rather than just a word? I know they are different units, but what's the evidence that the "prosodic word" is something different than what we think of as a "word" (at least in European languages)?
This is "the big question", whether (and why) to distinguish a Phonological Word = Prosodic Word (henceforth Pwd), from any other kind. Is it the same as a Morphological Word or a Syntactic Word? Indeed, is "word" the same in syntax and morphology? (Is syntax different from morphology?). Is the construct "word" actually necessary in syntax, morphology or phonology?
The weakest (most unmotivated) of these distinctions is a difference between word in the morphological versus syntactic senses. In the old days, there were rewrite rules that turned N into "cat, dog, cabbage", and ones that introduced affixes ("→ have+en"), and it was all about rules inserting things. Then some guy came up with the idea of generalized "lexical insertion" so that you don't have rules that rewrite N as "cabbage", but you do have rules that introduce grammatical affixes. The main impediment to a clear understanding of morphology-syntax relations was the murkiness of the theory of morpho-syntactic features. Rather than assigning a syntactic terminal a feature-like property (such as "subject marking") and letting a word-formation process cash that out as an affix, we had syntactic rules that actually manipulated affixes, which could hop. Focusing on theories of the 80's and 90's, the unresolved question is "what the heck is a word, in syntax, and what about morphology".
Clitics played a major role in this debate. There are very many cases where some word, especially a verb, has an affix that seems to serve a syntactic function so it "must" be a syntactic unit of its own, and yet it seems clear that the stuff is part of some other word. Object pronouns are a classic example, where in a number of languages, verbs are inflected for object in some way, with stuff that is "on" the verb (example: Lushootseed ɬušudxʷ "will see", ɬušudubicid "will see you", where the pronoun object triggers inflectional allomorphy – I do not think that "clitic" is the right analysis here, but this is a good illustration of the kind of fact that leads to "Pwd"). English possessive 's is another example. The basic problem is that you have an affix whose distribution suggests (for some theories) that the thing is really a syntactic terminal, like an "object", but by other standards (especially the "subject to allomorphy" test) it seems to act like a morpheme within something else (word, obviously, since morphology puts together words, not combinations of words)
A proffered solution (mid 80's vintage) to the general problem is to say that there is a word in syntax, where you state the sentence-level distribution of "words", and then sometimes two elements (especially clitics plus "real" words) are restructured into a phonological construct Pwd. The transition to this thinking does involve many more entities, so we had Pwds and Clitic Groups for a moment, but it was eventually concluded that we could dispose of some of that baggage, and simply have Pwds, which you get when "other stuff" like clitics are combined with real words.
(I pause to mention that there is a solution to the clitic problem in HPSG, promulgated by Phil Miller, which incorporates the notion of "edge feature", whereby a feature goes down some edge of a constituent. English possessive 's is an example: is end up at the right edge of the relevant NP, and does not care what kind of word it is realized on, and that is why you have possessive marking on verbs in phrases like "The man who I helped's house is burning".)
In phonology, people got enamoured of the idea of building structure on top of phonemes, so we liked the idea of a representational thing that corresponds to "word". It had to be a special phonological thing, given the (mistaken) axiom that phonology has no access to grammatical structure. Consequently, lots of kinds of structure was posited. Inkelas in her dissertation, for example, gives a representational reconstruction of Lexical Phonology level ordering, where the stuff that is "Level 1" is really stuff dominated by α, stuff that is "Level 2" is dominated by β, and so on. McCarthy & Prince in their account of Axininca Campa ("Prosodic Morphology II") profitably exploit the idea of manipulating Pwd boundaries, and people soon realized that you can have all sorts of constituent breaks inside a "word" where the edges are nudged this way and that to satisfy some constraint. The literature on Bantu is replete with numerous named constituents, like the Stem, Macrostem, Derivational Stem, Inflectional Stem and so on.
There grew to be an "enough is enough" feeling, there being a problem with so many constituents. Ito & Mester moved things in the right general direction, by basically getting rid of most of the structural distinctions, having just "Word" and "Foot" (and syllable and mora), but allowing these to be recursive – and in a laudible move, did not posit a bar-level distinction (W0, W', Wmax) as a thing in the representation. Instead, rules / constraints might mention that the things are "in the same minimal W", or that the W is "a top W" (kind of an A-over-A for phonology), and that's it. The evidence is pretty decent that you don't need a big inventory of structures, you just need recursion.
We understand now that "Phonological word" is, simply, "word", and there isn't some other sense of "word" out there. There is a recognition that the syntactic relationship between word and what might influence the form of a word is complicated. The syntactic need for a special phonological concept of word as existed in the 80's has pretty much gone away, in the face of Minimalism now accepting the concept of features, with Derivation by Phase, and nano-syntactic analysis (to the extent that these are accepted).
So yeah, there isn't any need for the distinction anymore.
A normal word is actually very hard to define. Do we write "dog house" or "doghouse"? (Or use a hyphen?) And of relevance to other languages more than English, do particles or clitics get written as separate words, or do they get joined together? Sometimes even affixes can be written as separate words.
A phonological word has a clear and relatively uncontroversial definition. (Though edge cases in many languages will of course be debated.) None of these orthographic conventions, derived over the centuries when the words in question may not even have been pronounced the same, are relevant.
(And I think, though I'm not certain, that clitics, though they fit in between words and affixes from a morphosynactic perspective are just simple parts of phonological words.)