I'd say 要 is more direct and affirmative, like "I've made up my mind and this is what I want." Where 想 is more like casual, "would like". 想 can also mean to think about something, or to miss someone. Hope that helps give an idea of the feeling.
In another answer previously it said wo yao chu zhongguo and the answer was I need to go to China -- this one is I want to instead. It is a more literal translation but inconsistent. Why was it needs to before and want to was wrong??