Right to vote was already established before the change of the name (subject->citizen).
So, what changed? Well subjects have “privileges” that are afforded from the monarch, and citizens have “rights” which are given from the state.
Except:
1) In olde english law, the monarch and the state are literally the same thing.
2) Rights seem to be pretty loosely followed if they’re actually, you know, RIGHTS, and not privileges afforded from the state.
I’d say that semantically the difference is how the words make you feel, not the actual applicability of the terms to anything that has been realised.