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I'm a native speaker of a language which has syllabic consonants, here are the examples
კლდე/k'lde, [kʼl̩dɛ] "cliff"
ქრთამი/krtami, [kʰɾ̩tʰami] "bribe"
ბრძენი/brdzeni, [bɾ̩d͡zɛni] "Wise"
And to my ears they all sound like [kʼᵊldɛ] [kʰᵊɾtʰami] and [bᵊɾd͡zɛni] respectively.
What's difference between a syllabic consonant and əC?
There are two answers, one in phonology, the other in phonetics.
The phonology answer is at least initially mildly stupid: it says that [mr̩t] is a syllable with three segments, [r̩] being the peak and neither the onset nor the coda having two segments, but [mərt] is a syllable with four segments, [ə] being the peak and the coda having two segments, [rt]. You didn't ask about the third option, [mrt], so I won't get into that. This becomes a little less trivial when you look for evidence in a language that might support one of the other analyses.
The most important fact that could point to a difference is the existence of a contrast. This sort of exists in Sanskrit, where there could be two separate words वृक = [vr̩kə] versus वर्क [vərkə] – standard transliterational short "a" is generally recognized as being pronounced "ə". There could be aspects of the rule system that point to one analysis over another, for example you might argue for a syllabic r in [mr̩t] if the language otherwise had no two-consonant codas.
However, none of those considerations apply to Georgian. The language has gigantic consonant clusters. The one argument that might be advanced in favor of "syllabic consonant" and against schwa-plus-consonant is that there "is" no schwa in Georgian. The immediate retort is that there also "are" no syllabic consonants in the language. Neither of these properties are fully-contrastive and phonological. So the phonological answer is "none of the above".
The phonetic answer is based on physical properties of utterances, and not abstract analysis – there is a problematic semi-circularity in sorting out the phonetics vs. phonology of a language (let's not try to solve it right now). The difference would be seen (seen not necessarily heard) in the fine details after the release of the consonant [kʰ], for instance. You would look for a portion of the signal that has clear voicing and vocalic formant structure, where F1=500, F2=1500 and F3=2500 (approximately), and that would be evidence for a vowel schwa. If you just have trilling noise, that would be evidence for a syllabic (or non-syllabic) rhotic consonant.
To see this, you would take some example recordings (good recordings, the online examples that I found on Forvo were generally poor quality recordings) and subject them to an acoustic analysis using a tool like Praat. An alternative relies on just hearing, but it really requires appropriate ear-training as was traditionally given at Edinburgh, UCL and in the olden days Michigan. People still make ear-based claims, but then one should ask what their basis is for the claim, as opposed to an alternative.
You might want to read the linguistic literature covering Tashlhit, where the question of "syllabic" consonants is addressed both positively and negatively, where you can see how linguists address this issue. I'm not persuaded by either side, but the discussion is somewhat enlightening.
