The question that exists in phonological theory regarding syllables, moras and so forth is, what is the required collection of suprasegmental units required to describe human language phonological grammars. Practically speaking, this means, "do we need all of the set {skeletal position, mora, onset, rhyme, nucleus, coda, margin, syllable}?"
The literature is full of bits of evidence for each of these objects, what has not been undertaken yet is a systematic re-evaluation and purge of unnecessary devices. Let us simplify the theory just a bit to "skeletal position, syllable, mora, onset, coda" (skeletal position is the most vulnerable of theses); eventually we want to focus on the question "What thing can you do with a syllable that you can't do with these other devices?". And example of a difference between moras and syllable is "third syllable" refers to one thing, and "third mora" refers to different thing, just in case you have a language with syllables having more than one mora. "Third syllable" is called for if you must unify {cvcvcvcvcv, cvvcvcvcvcv, cvcvvcvcvcv}, and "third mora" is called for if you must unify {cvcvcvcvcv, cvvcvcvcvcv, cvcvvcvcvcv, cvvcvvcvcvcv}. If you need both kinds of expression, you apparently need both devices. However, there has been a general dogma (incorrect IMO but that's beside the point) that rules never seek "the third" anything, so there is no generally accepted example that points to syllables vs. moras of this type.
Another possible distinguishing feature would be to look for constituency evidence, in the form of rules that apply between X and Y as long as they are "in the same syllable". Some statements of the distribution of emphasis (pharyngealization) in Arabic are stated in terms of "being in the same syllable", but the empirical underpinnings of the Arabic claim are rather unclear. Crucially, "in the same syllable" has to be different from "adjacent to a vowel". Turkish is claimed, in an analysis by Clements & Sezer, to have a relevant rule where velars are fronted when they are "in the same syllable with a front vowel", and in words like /renk/ they are not adjacent to the front vowel (I can neither confirm nor deny the truthiness of the data claim).
The program of defending the syllable therefore reduces to accumlating all of the evidence offered in support of the syllable, and re-considering whether some other device is sufficient. As an example, Kahn in his dissertation argues for the syllable as necessary for explaining English consonant allophony, but an alternative program of stating the generalizations in terms of "foot" has been rather successful. The theory of "Radical CV phonology" has only the prososodic objects C and V, period, and it turns out to be able to state the majority of the relevant generaliations well enough (however, it is more abstract, in the sense that C does not always correspond to a phonetic consonant and V is not always a phonetic vowel).
