To answer "Tu veux venir à la piscine?", is there any difference from saying "Oui, je le veux" instead of "Oui, ce que je veux"?

Fergus,

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.