More or less, in practice your US broker has to respect "best execution", meaning it cannot offer you a price worst than the "NBBO", which is the composite of all US stock markets.
In practice though, I expect most companies listed in NYSE Texas to be solely listed there.
> “normal NYSE” vs “NYSE Texas”
There's already _a lot_ of "NYSE" markets and segments. Some exist because of historical mergers (Amex, Arca), and some to offer alternate / cheaper listings (NYSE listing is very expensive vs say Nasdaq)
> What benefits would companies see from listing on NYSE Texas vs the other NYSE?
Unclear at the moment but I expect a lower bar for listing requirements and a cheaper price. Nyse is by far the most expensive listing there is, but it's also very exhaustive on listing requirements (audits, etc). I guess there is also a political/economical deal with Texas at play, to incentivise companies to move there, list there, and grab some of NY market share.