You have given us no contexts, so we cannot judge them.
'Get' means 'acquire, obtain'. 'Buy' means 'acquire with money'. So buying is just one method of getting something. However, we presume that the father bought the car.
"My dad just bought me a new car" is specific and states that your dad paid for the car so that you now have it.
"My dad just got me a new car" has the same tangible conclusion (a new car) but it leaves open the possibility that your dad might have bought it, traded for it without buying, won it in a contest, or inherited it. Or maybe he owns the factory and just decided to
Thank you so much both you guys. Does it sound too ambiguous, if you use 'get' even when you know that your dad bought it? Do native speakers strictly distinguish and use one or the other for appropreate situations?