He would buy a new car is likely to be used in a context in which such a purchase had turned out to be ill-advised. Such a sentence might, for example, be followed by I knew he couldn’t afford it, but he wouldn’t listen. In practice, the sentence would probably be He would go and buy a new car. Additionally, it might express a habitual past action: He would buy a new car every year even though he didn’t need one.
He said he would buy a new car is reported speech, but it is not possible to tell from that sentence alone whether he had bought a new car or not. It could be followed by . . . and so he did (so he’s bought one), or it could be followed by but he changed his mind (so he hasn’t).