Omnisexuality is defined as an attraction to persons of any gender identity (incl non-binary), but gender does still matter for an omnisexual (so in my case for example, I'm attracted to everyone, but more strongly towards men and less towards non-men). Bisexuals are attracted to men and women, but could also be attracted to non-binary people (but not necessarily I think).
So I'd say there's very little difference between them, if any. It's more of a vibe-based thing, whatever label you like the most.
Bisexuals are not always attracted to men and women. A bisexual is simply someone that is attracted to two or more genders. That typically includes men and women, but doesn't have to.
I've heard of bi as experiencing both homosexual (same-gender) and heterosexual (different-gender) attraction. That's the definition I like, as it's not necessarily non-binary exclusionary and so accurately describes how I experience attraction.