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.

I hear you! As I learned Korean by myself, I had the same dillema for a long time :)). I think the answer to your question is that both 됐어 and 되었어 mean the same thing. Only that the 됐어 is the shorter form of 되었어. (I guess all languages tend to have this kind of contracted forms, English including. I'd call this tendency "speaking economy" or.. "the economization of speaking" :)))~)