Brand-new means entirely new, that has not previously been used. See freedictionary, Webster, Macmillan.
A brand-new car is a car that's just been bought from the shop and is not supposed to have had previous owners.
A new car means the owner hasn’t had it for a long time, but it doesn't mean the car hasn't had any previous owners, it could be second-hand.
Saying "my car" doesn't give any indication on the age of the car. He could have had it for years.
Obviously the speaker isn't going very well, we can expect he is very cross because we don't expect a car we have just bought as new from the shop to be "a piece of junk", which means it's in bad condition, low quality.