Yes I think the former is more general/everyday and the second is rather more emphatic.
I think the second should be written:
"Oui, c'est ce que je veux !" - Yes, that's what I want (to do) !
Oui, je le veux. -- Yes, I want it.
Oui, ce que je veux sounds wrong to me (Yes, what I want). I would say Oui, c'est ce que je veux. (Yes, it's what I want.)
In this case you could, I believe, just say: Oui, je veux.
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