I've heard from some rabbonim and educated laymen that there are some differences between Chabad mikvaos and the more standard ones. This makes it so Lubavitch women can't use any other mikva. Is there a real difference or is it a political thing?
It's not politics (at least not purely); it's a matter of construction. Suppose you have a big pit in the ground that collects rainwater; everyone agrees that pit is a mikvah. Fine. Today, people want a mikva with mostly clean, heated water. So they immerse in a pit of clean, heated water that is connected via a small opening to the rainwater pit. (Or usually there's an intermediate pit in between, to keep the for-dipping mikva the cleanest and warmest.) How do you connect the two pits?
Chabad builds bor al gabei bor, "pit on top of pit." If you immerse in a Chabad mikvah, you'll look down at the floor and see a small hole which leads to another pit of more-rainwater (and then another one after that). Others build them bor al yad bor, "pit next to pit." If you immerse in a non-Chabad mikvah, look along one of the side walls and you should see a hole leading to another pit.
So there's a legitimate halachic debate on which is the better design. Obviously ask your rabbi, but non-Chabad people would have a strong halachic preference to follow a mikvah that their Achronim say is the right design.
This article explains the halachic pros and cons of the Chabad mikva.