Although both seek for the same question: Where do you come from? I like to know the precise context of these two question phrases in Italian. Also where we use "Di" and "Da" specifically?
As others mentioned, the two expressions you posted don't mean "What's your name", but rather "Where are you from?". If you wanted to be literal in the translation, you would translate "Da dove vieni?" as "Where do you come from?".
Most of the time, both expressions are used to mean exactly the same thing, so it doesn't really matter. "Da dove vieni", however, can be used to ask what should instead be "Da dove sei venuto?", which if you want to keep the meaning should be translated as "From where did you leave to come here?". Though realistically it would be better to ask "How did you get here?" ("Come sei arrivato qui?").
To sum up, the meaning is almost the same. There is the nuance outlined by user9079251 as well as the double meaning I pointed out, but more often than not they can be used interchangeably.
Actually "Di dove sei?" means "Where do you live?" and "Da dove vieni?" means "Where were you born?" or "Where do you come from?". To ask a people "where do you live?", you can use also "Dove abiti?", which is more formal.