What is the Difference between 되었어요 and 됐어요? For example: -서울에 산 지 얼마나 되었어요? -오 년 됐어요. Or -일 년 육 개월 정도 되었어요. I know both answers are correct.
I have always considered 됐어요 to be a contraction of 되었어요. They are both equally correct forms of the past tense of the verb 되다.
In spoken Korean, I use 됐어요 almost exclusively. In written Korean, I am more prone to use 되었다. Although there is no hard or fast rule on why this is, it is what feels most "correct."
Just like in English, contractions are sometimes considered less formal. But they certainly are acceptable, especially in spoken English. This example with 되었다 versus 됐다 is quite similar I feel.
I want to add some more to Athony's answer.
I would say there is no difference between 됐어요 and 되었어요 in conversation as they sound just identical if you speak quick enough. If I can imagine someone confessing his/her crime to Police, he/she may say in a very dark mood "의도한 것은 아니지만.... 그렇게.... 되었어요" Or, a teacher telling fairy tale to children may say "개구리가 멋진 왕자님이 되었어요". In daily usage I see no distinction between two.
However, 됐어요 and 되었어요 are clearly different in their politeness when written. I would write 됐어요 to my parents or 형/누나; 되었어요 to my close teachers / professors / senior colleagues. To someone I haven' met before, or in business, I use 되었습니다.
To be more specific, I never use 됐어요 to my supervisor, but I occasionally use 됐어요 to another professor who is almost a friend of mine. Another thing is that 되었습니다/됐습니다(spoken) sound more formal than 되었어요. 되었어요 do sound polite, but it almost sounds cute in a way.
Adding to @Vladhagen's great answer, there are a few more things that you need to understand when learning Korean.
The longer word or phrase (sentence) you use, the more polite and honorific it becomes.
It doesn't necessarily mean "됐어요" is an impolite and non-honorific form. It is. However, there is subtle difference and it is so subtle that it is very difficult to tell.
이번 여름에 유학을 가게 됐어요. (three blocks) I will go abroad to study this Summer.
이번 여름에 유학을 가게 되었어요. (four blocks)
이번 여름에 유학을 가게 되었습니다. (five blocks)
이번 여름에 유학을 가게 되었사옵니다. (six blocks. This expression is the most honorific form that was used to a king or people in extremely high position. You don't use it nowadays unless you want to make a joke)
No. 2 is used when you speak to people you feel less familiar and friendly than those to whom you would use No. 1. No. 3 is the most honorific sentence among the three.
"됐어요" and "되었어요" are almost exclusively used in conversation.