In France, we actually teach children to use "voudrais" instead of "veux" as it is more polite.
"Maman, je veux un gâteau!
- Non, on dit 'je voudrais / j'aimerais un gâteau, s'il-te-plaît maman' "
:)

Bonjour Darren ! Je veux means "I want" - it's very direct. Je voudrais means "I would like" - it's softer and politer, so you'd use it when ordering in a restaurant or shopping, for example.