First, You must know there is polite level of Korean. 뭐해? (low level polite) 뭐해요?(middle level polite) 뭐하십니까?(High level polite) Second 뭐하는 거에요? VS 뭐해요? 어디 가는 거에요? VS 어디가요? (Above expression is all middle polite level. You can see "요" the end of all sentences.) I think meaning is exactly same. The nuances and tones are almost same. The *a few* different is the former is *a very little* more polite than the latter. Third 뭐하는 중이에요? VS 뭐해요? 어디가는 중이에요? VS 어디가요? You can see new word "중". "~중이다" mean the present progressive form. For example, 나는 TV를 보는 중이다. I'm watching the TV. 나는 TV를 본다. I watch the TV. 나는 TV를 봤다. I watched the TV. 지금 뭐 하는 중이에요? What are you doing now? 지금 뭐해요? What do you do now? 어디 가는 중이에요? Where are you going? 어디가요? Where do you go? Have a good day~

Short forms always work in spoken as other languages do, and I think "~ 하는 거예요?" is often found in textbooks to explain how the Korean language works in terms of grammar. This kind of the style of the speech is more prefered by women, I think. In daily conversation, "뭐하세요?","뭐해요?" and "뭐해?" are often used. It's my two cents.^^ \^o^/

I asked about this a while ago and i think my answers were a little bit better. Take a look: http://www.italki.com/answers/question/93276.htm

The difference is degree of uncertainty or strength of will . 나는 공원에 갈거예요.-I will go to the park. (maybe) 나는 공원에 가요.- I will go to the park (without doubt)

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