The difference lies in the level of politeness. Using "feel" in both cases seems more tentative and hence more polite.

It you 'want' something it is very definite. Maybe you don't want to go for a walk in the forest because you know that it is dangerous. Or maybe you don't want to go for a walk because you need to stay home and study. Or maybe you just hate walking! Small children, especially spoilt ones, say 'I want ..' and 'I don't want' a lot, because they have definite preferences and haven't yet learnt to be polite. 'I feel like' is much less definite, and much more temporary. It's what your body is telling you at the moment. Maybe you like going for walks usually, but today you're feeling a bit tired and lazy, so you say 'I don't feel like going for walk today. Can we go tomorrow?' NB I've used the British English 'go for a walk' rather than the American 'take a walk' here, but the construction and meaning is the same. I hope that answers your question.