The first thing to note is that the cu- and quo- spellings are equivalent: quoius is an earlier form of cuius, in both the functions you describe. I think your question is really about the relationship between:
- the gen. sg. of the relative pronoun, written quoius in Old Latin but cuius in Classical Latin; and
- the possessive adjective meaning 'whose', written quoius, -a, -um in Old Latin but cuius, -a, -um in Classical Latin.
Note that the second form is not exclusively Plautine, but survived into Classical Latin: we have cuium pecus in Vergil, Eclogues 3.1, a couple of uses of cuium and cuia in Cicero, and a few other scattered instances.
The relationship between the two is not entirely clear. The most obvious explanation (thus de Vaan) is that the possessive adjective was derived from the gen. sg.: cuius was reanalyzed as the nominative singular masculine form of an adjective, and a new paradigm was created.
However, it's also possible that both forms were inherited from Proto-Indo-European, since there's a Greek adjective ποῖος 'of what kind?' which looks formally equivalent to cuius, -a, -um and could be a cognate.
Your last question -- why the adjective disappeared -- is of the kind that historical linguistics finds hardest to answer. Forms fall out of use in every language often for no apparent reason, and there are many other Plautine usages which disappeared in later Latin. In this case it seems plausible that, since the possessive adjective had the same function as the genitive of the relative pronoun, there was no need for both forms and the simpler form (the one without an entire paradigm) won out.
ETA: though maybe it didn't disappear: Spanish has a possessive adjective cuyo, -a 'whose'. Whether this is a survival of the Plautine usage or an independent later innovation, I don't know.