Yes, and no. Not in the sense that it's one uniform trading system, but there are various interlinkages (market data via the SIP for example; mostly dictated by RegNMS) and most of the exchanges operate a brokerage running an order routing business.
> Would there be any difference for me buying and selling stocks on “normal NYSE” vs “NYSE Texas”?
Assuming you have the ability to dictate to your broker where they perform the trade there would be absolutely no difference between trading on NYSE vs NYSE TX (minus maybe some currently undecided fee differences). Functionally they are identical (they even run on the same technology stack).
> What benefits would companies see from listing on NYSE Texas vs the other NYSE?
The cultural issue TXSE (and by extension this NYSE TX move) are trying to capture is the 'anti-DEI' / 'the exchange tells us what the composition of our board must be to meet listings standards' type things. There's a subset of the corporate world who see value in capitalizing on these issues. There's also the potential for different financial requirements or incorporation requirements, but those haven't been disclosed yet (and wouldn't be too divergent from the existing differences between listing on the various Nasdaq or NYSE exchanges).