'David finally sold his old car' is in the past simple, and it refers to something that happened in the PAST. He might have sold it yesterday, six months or ten years ago, but we know that this is a past event. There is no relationship with the present time.
'David has finally sold his old car' is in the present perfect, and it relates to the PRESENT time. We know that this is a recent event, with an impact on the present moment. The key to understanding the use of the present perfect is realising that a present perfect sentence always gives you information about the current situation. If you say this, you are telling the listener something about now. Perhaps David is really happy because he's sold his car, or perhaps he's finally got some money and can pay his debts. You might say 'Hey, David's finally sold his old car. Great! At last he can give me back the money he owes me. I'll phone him up now and ask for it'. Or you might say, 'Hey, David's finally sold his old car. Great! Now I can park on our shared driveway.'
Note the word order, by the way. The correct word order is 'David has finally sold his old car' . Remember that adverbs generally go between the auxiliary and the main verb.