'I want to xx' is a neutral statement. If you say 'I do want to xx' you are adding emphasis to your statement, often to contradict a suggestion that it isn't true. For example: We're going for a picnic tomorrow. Are you coming? Oh, sorry, I can't. Why don't you want to come? It'll be fun. I do want to come, but I've just got too much work to do.

Yes, there is a difference. "I want to" is common. It is straightforward. It means exactly what it says. It is the form that should be used by people who are learning English. "I do want to" is less common and carries two possible extra meanings. 1) Intensification, expressing a stronger feeling. These three sentences express three increasingly strong feelings: a) I want to see the total solar eclipse in 2017. b) I do want to see the total solar eclipse in 2017. c) Oh, I do so want to see the total solar eclipse in 2017, I'll be heartbroken if it's cloudy. 2) Responding to an earlier statement--answering a question, confirming or contradicting someone's guess about your wishes. "We could go out to eat at La Sarape, if you like Mexican food." "I do like Mexican food."