Come in native english speaker, I have something to ask

There is a difference between "wouldn't" and "couldn't" but in this particular sentence, it is not easy to distinguish. Let's try two other sentences that I think will be easier to understand. 1) The babysitter told the little boy to tie his shoes but he couldn't because he is very young and doesn't know how. 2) The babysitter told the older boy to tie his shoes but he wouldn't because he doesn't like being bossed around. In sentence 1, the boy couldn't do something he because he didn't know how. In other words, he was not able. In sentence 2, the boy wouldn't do something because he decided not to do it even though he was able. Getting back to the car question, I am not an English teacher, but I think "couldn't" would be a better choice. Either way, people would understand what you are saying.

Both sentences do mean that your car failed to start, but the latter makes more sense. Saying that your car "wouldn't" start implies that the car had a choice in the matter and decided not to start, while "couldn't" means that there was something wrong with the car that prevented it from starting. Or, as Transformed said, you could always just say "My car didn't start" and be done with it.

If a car "would not" start meant that it didn't start for whatever reason. If a car "could not" start, it is because of some incapability of the vehicle. They both do mean that the car "did not" start that day. Semantics in a sentence can very much so determine how a sentence is interpreted.

I think there's no difference in meaning. They are both saying that the car didn't start.

The meaning of the first sentence is that you are predicting with or without having tried that the car is not going to start,and the later is like the car tried to get started but failed.