Citizen rights for the whole federation are the same. You can move free, no division. Difference is in budget distribution and type of local government elections.
In general citizens have no restrictions. Just have additional rights like to elect current republic president. Although Moscow and St.Petersburg have special laws to work and live there. You should have a permission as far as I know. But it is for any citizen, doesn't depend on federal subject.

For example Presidents of the republics are elected by locals (usual voting of every citizen). Federal region heads are settled by the federal government. As I said every republic has its own relations level with the federal government.

Not sure about funding for schools and etc. but it seems that it is discussed at the lower level. Cities, etc.