Selling vs a dealer trade-in
The honest framing
A trade-in is the path of least resistance: you hand the dealer your old car, they knock its agreed value off the price of the new one, and you drive away having dealt with a single business on a single day. There is no advertising, no separate handover, and often a paperwork and stamp-duty convenience when the trade value reduces the price of the car you are buying. For a lot of people that simplicity is the whole appeal, and it is a genuine one.
The trade-off is that a dealer is buying your old car to resell it at a margin, and the trade figure is usually wrapped inside a larger negotiation where it is easy to lose track of what each part is really worth. Selling your old car separately to a direct buyer breaks that bundle apart: the old car is valued on its own, and you negotiate the new car on its own. The cost is a little more coordination, because the two transactions are no longer one. This page lays out that trade honestly so you can decide which matters more for your situation.