Changing airports is almost never something you planned. Berlin Brandenburg Airport to Stuttgart Airport is 656 km and about 6h 28, and a car covers it in one leg with no change and no ticket barrier between you and your suitcase.
The pickup is driven by your inbound flight number: the driver watches the actual landing time and waits 60 minutes free after touchdown, which is what a diverted or delayed arrival needs. At the far end the drop-off is at the correct terminal for your departure — Stuttgart Airport has 4 and the wrong one costs a shuttle ride you do not have time for. Tell us the departing flight at booking and the schedule is worked backwards from its check-in deadline rather than forwards from a convenient departure. We run the same journey to Munich Airport, priced the same way.
Luggage stays in the car for the whole 6h 28, which is the single biggest practical difference from any rail route: nothing is lifted onto a train at Berlin Brandenburg Airport, off it again at the interchange, or onto a second one for the run into Stuttgart Airport. If the airline has rebooked you and is reimbursing ground transport, the invoice shows the route, the date and the fare in a form an airline will accept.
What you are committing to at 6h 28
If you have read the comparison above and still want the car, this is what you are committing to. One driver and one vehicle for the whole distance — no handover, no change, no platform — and a comfort stop wherever you ask for one, which on this length most people use once. The reason people choose it at this distance is almost never comfort. It is that across 6h 28 no gate closes on you and there is no second boarding pass to lose: the deadline you are driving towards is the only one left in the day. Your luggage is handled exactly twice, once at Berlin Brandenburg Airport and once at Stuttgart Airport, and the rebooking calls you still have to make are easier from a back seat than from a platform.