"Prosody" derives from προσῳδία, which you can read all about here: it roughly is about pitch being superimposed on speech. In Latin this was extended to poetic meter. Modern linguistics (in the past 70 or so years) has extended that to pretty much all properties. It is not a technical concept, so it doesn't have a precise definition and it doesn't sharply contrast with something else. The essence of "prosody" is that it is any sound property of speech that can extend over more than one segment. It is equivalent to "autosegment".
"Segment" on the other hand is a much crisper concept, although there are unresolved debates of the "one segment or two?" variety, e.g. "Is [tʃ] one segment or two?" – typically the answer is deferred to some higher-level a priori postulate such as objecting that [tʃ] can't be a cluster because in the language there aren't any other obstruent clusters in the onset. In feature-geometric theories the answer is trivial: something is suprasegmental iff the thing dominates the root node (mother node of all segments).
The London School of Phonology spearheaded by Firth would give a "prosodic" analysis to whatever it could, and that use of "prosodic" is very similar to "autosegmental". In essence, vowel harmony would be said to be prosodic just in case it marks off a continuous span of speech. By inspecting the history of vowel harmony in autosegmental phonology, one can clearly see that "suprasegmental" vs. "segmental" is completely orthogonal to dominance relations. In early work on harmony, harmonizing features such as front/back, round, ATR (see Clements' Vowel harmony in nonlinear generative phonology) were "suprasegmentalized" just in case they spread. However since the first hierarchical model of features in Clements 1985, it was understood that sub-segmental features can spread just as well as suprasegmental features.
It is completely controversial where tone features in Chinese (any Chinese language) attach -- subsegmentally or suprasegmentally. Sticking to the most-Greek understanding of "prosody", only tone would be prosody, so everything else would be non-prosodic: therefore, Chinese at least has been analyzed as a segmental prosodic feature, and Turkish (in Clements' analysis) has been analyzed as suprasegmental non-prosody.
However: that is all just a terminology game. There is no utility to the term "prosody", which has largely been coopted by people working on intonation. The distinction "subsegmental" vs "suprasegmental" refers to something well-defined which is not easily detectable (that is, we know what the terms mean, and we don't have tools that easily identify whether is it one versus the other), for which reason "suprasegmental" is a fairly useless term as well.