Do you guys change address on aadhar when moving out to a rented apartment? I'm not sure about any advantages of changing the address on aadhar everytime you change the home (specifically talking about rented accomodation, not buying a home), are there any?

Not much of an issue. You can give a rent agreement as a proof of residence along with aadhar for most purposes.

I do. It makes things much easier for later when someone asks for an address proof and doesn't accept rental agreements.