This is one of those areas in which the IPA (and its use by transcribers) gets a bit messy. The IPA transcription of an utterance is generally meant to be the phonetic representation of that utterance (leaving aside "phonemic transcriptions", which usually borrow symbols from the IPA set out of convenience), but very often the assumed phonology for the language at hand is projected onto the phonetic transcription. This tendency is seen, for example, when a transcriber provides a "broad" phonetic transcription that abstracts away from the aspirated/unaspirated contrast because the language in question does not have a phonological aspiration contrast. The IPA even provides a set of diacritics for phonological phenomena like stress and tone.
The syllable and syllabicity are generally phonological constructs. Please see this related post:
Is syllable a phonetic or a phonological concept?
As @jlawler points out in his comment there, there are some phonetic (articulatory) models that include a phonetic notion of a "syllable", but more often than not, when a transcription includes information about syllabicity and syllable boundaries, the transcriber is imposing her assumptions about the phonology of the language on the transcription.
Leaving the syllabic trill aside for a moment, let's think about a (presumably) more familiar example. What is the difference between [u] and [w]? Well, it depends on the language. What about in English? In many cases, there is no difference, articulatorily or acoustically speaking. In spectrograms, the two can look identical, depending on the context, and the tongue and lips can be positioned in the same place for both sounds. So why are they given different symbols and different names (vowel vs. approximant)? Because in the phonology of English (and of French, which, together with English, formed the basis for much of the IPA), this sound can function as a vowel (i.e. the nucleus of a syllable) or as a consonant.
So, whether a sound in question is transcribed as [u] or as [w] does not give you any absolute information about how it is articulated. However, because of some well-known, generally agreed-upon principles, such as the tendency for sounds transcribed as vowels to be longer in duration than there approximant counterparts, given the same rate of speech, it does give you some information in relative terms--much in the same way that stress marks or tone marks convey information about the syllables with which they are associated relative to other syllables. And when [u] and [w] appear in the same syllable, like in the English word woo, as a phonetician I know to expect the vocal tract to start more constricted in the [w] than it ends up in the [u] (resulting in higher F1 and F2 in the [u] than in the [w]), and for the duration and intensity of the [u] to be greater than those of the [w].
Circling back to the syllabic alveolar trill and the (non-syllabic) alveolar trill, this pair of sounds in the relevant Indian languages is analogous to the [u] and the [w] in English. The "syllabic" version of the sound is used in transcriptions when the transcriber assumes it to be functioning as a syllable nucleus, and the "non-syllabic" version is used in other cases. As other responders have noted, there is nothing fundamentally different about how the two sounds are articulated, except in relative terms in some contexts.
