됐습니다 is the shortened version of 되었습니다. You can freely alternate between the two forms without any change to the meaning.
The only exception is when 됐습니다 is being used to express dismissal, as in the case below:
-A : 물 더 드릴까요?
-B : 아뇨, 됐습니다.
In this case, 되었습니다 sounds wrong. I think the term 됐습니다 is in the process of being idiomatized here.
So, to sum it up, 되었습니다 and 됐습니다 are the same, except only the form "됐습니다" can be used as the idiomatic term of dismissal.