Well, there's a little difference. "Autó" refers to "car". But for example horse-drawn carriage is "lovas kocsi", and we don't use autó to describe it. And transport equipments that we use with our hands are also can be called "kocsi". Hand trucks, etc.
As I bet you guessed, autó is reference to a car being self-propelled as opposed to a lovas kocsi that is pulled by a horse and other types of "kocsi" pulled by hands etc. As an interesting (at least to me) side note, kocsi literally means from Kocs which was the city where some sort of original carriage was invented. This is why some form of the word kocs shows up in a lot of languages. Even English uses coach sometimes although carriage and cart have become a bit more common.
Yeah, this is Hungary. :D And Kocs is famous for its milk, too. 10 years ago I drank "Kocsi" milk.
So if "a" and "az" both mean "the", is the second one used when the word after it starts with a vowel?
autó is the foreign word for this vehicle, kocsi (--> coach) is of Hungarian origin https://en.wikipedia.org/wiki/Coach_%28carriage%29
The "a" in "az" sounds more like a schwa... is there any rule of thumb to follow for when a is pronounced like this (not sure if it's a true schwa) or closer to actual [a]?
My experience with Hungarian is this is not a true schwa sound, it is because it is spoken quickly in between the "és" and the "a" of autó. Slowly and carefully would make the "a" the same whether "a" or "az", which is not like any other European language "a" sound that I am familiar with.