I live in Georgia, a Right-to-Work state, and everything MatthewShiner said is pretty accurate. I'm not Equity, there's not enough AEA work in this town to make it a worthwhile choice. But I did a lot of work at one of the few Equity houses in town, and when it was on Equity shows, everything"acted" like I was Equity. I got weeks toward getting my card if I wanted it, the theatre had to pay into pension and health weeks, they had to pay me at least Equity minimums and all other Equity rules were in place for myself and the cast in regards to breaks, overtime, extra duties, etc.
If you're already Equity, I don't think things in the day-to day experience should change for you. If you're not Equity and you get on an "Equity" show, it will be a big education! I'm glad I got the opportunity to learn so much about how a show functions under union rules before actually having to make the decision to join or not.