This usage of "would" is a little tricky. When you are at the wheel trying unsuccessfully to start the car, you normally say : "the car will not (won't) start". You could also say : "the car is not starting", but the usage of "will" transmits a feeling that the car is refusing to start and that you are frustrated by it. When you report this event in the past, you change the "will" to "would". e.g. My wife will not tidy the lounge, no matter how often I ask her. (reporting the past) I asked my wife to tidy the lounge yesterday but she wouldn't (do it). Like you said, you could also say "the car didn't start", which sounds more factual and neutral.

I want to emphasize that the difference in meaning between "My car wouldn't start" and "My car didn't start" is VERY SMALL. In real life, any native speaker might use either one. It expresses the speaker's feeling about where to place the responsibility! In real life there would be more to the story that would make the full meaning clear. As others have said, "didn't" simply states a fact. "Wouldn't" suggests that it is the car's fault! I did everything right, I tried and tried, but that disobedient car refused to start!" And a third possibility is, "I couldn't start the car." That puts the responsibility on me! "My wife says you just need to jiggle the key a bit, and she is usually able to start the car, but I couldn't start the car."

When you say "the car didn't start", that is a statement of fact, nothing more to it. This is the most neutral way of saying it. When you say "the car wouldn't start", then there is a presumption underlying the statement that the car has done this before, doesn't start regularly. If you are talking about people, there is also an additional possibility when "wouldn't" is used: "He wouldn't come home on time" implies that he didn't want to come home.