The difference is literally using UDP over TCP. The problem QUIC tries to solve is that many of TCP's design decisions were made in a different environment that the one we live today (eg. being able to switch between from 2 internet sources like Wi-Fi and mobile data while retaining connections). Plus, it integrates TLS in it to reduce protocol overhead.

IIRC even UDP isn't ideal, the reason for choosing it over making a brand new protocol was to avoid network devices like routers dropping packets that they wouldn't recognize.

WebSockets is just an abstraction over a reliable protocol (TCP), therefore, WebSockets could theoretically work over QUIC too.