The problem with the question is that it simple depends. So first, even within the roman empire there was change. A Roman from 500 b.c. would not think 500 a.d. was very roman. A roman from 500 a.d. would not think 1400 a.d. is very roman.
So the answer to your question really is, in the begginning they where the same and then slowly shifted away from it. What can be said is that there is no clear break, no point where we can clearly say "here" now its something diffrent. This is why we have so many problems in nailing down the end of the roman empire or the begginning of the byzantine empire.
So I think, the only sencable thing to do is to still call them romans (or rather the roman empire).
So the answer to your question really is, in the begginning they where the same and then slowly shifted away from it. What can be said is that there is no clear break, no point where we can clearly say "here" now its something diffrent. This is why we have so many problems in nailing down the end of the roman empire or the begginning of the byzantine empire.
So I think, the only sencable thing to do is to still call them romans (or rather the roman empire).