Yes, both sentences are correct.
There is very little difference. "To start" and "to get started" are quite often used synonymously.
The only difference I can think of is that "do you want to start" places added emphasis on the person who is going to start, as opposed to the activity or action that is being started itself. E.g.:
Do you want to get started? (= Should we start doing this?)
Do you want to start? (= Do you want to be the one who starts? Or do you want me to start instead?)
"To get started" also seems to imply that whatever is being started is a longer, previously planned activity, whereas "to start" is slightly more generic and could refer to any arbitrary action. This, of course, is not a rule.
Finally, there are specific situations where "to start" seems to be a more suitable, idiomatic choice (if you were starting a speech, for instance). E.g.:
"I would like to start by thanking you all for being here..."
I hope this helps.