There's not much difference at the end, except that you can consider that the second way of doing things will let you still use the old car, only it'll be new.
However, the car of Theseus example allows for a worrisome wrinkle. What if you kept all the old parts during the process you describe, and then after, you re-assembled them? Which car has the superior claim to being identical with the old car? I think it obvious that the reassembled old parts have the superior claim over the car produced by part-by-part substitution of new parts.
Yes, it has all the parts of the old car, and only those pieces, but it was assembled anew from zero.
A more difficult dichotomy would be: what if we exchange parts between two cars until each one is the other one. That's hard to decide.
Or how about we make all the new parts plastic so that no functioning "car" is produced by the gradual part-for-part substitution. We end up with two cars, one an all-plastic car model, and one consisting of all the old parts assembled and functioning. Which has the better claim to being the old car?
Yeah, part exchanges quickly leave me with even less salient intuitions about the proper use of old identity terms than these more "splitting" oriented cases. What should we say if a lion talked and all that.