At the time

You really work on stumping us with your questions don't you? (Just kidding )
There is not much difference, but I'll let someone else try to explain it.

The difference is rhetorical, at that time sounds slightly more grandiose, but the meaning is the same.

Yeah. I get it.

"I would have put a V8 in my car, but at that time I couldn't afford it."
(Separates that time from this and implies that now I can afford it)

"I would have put a V8 in my car, but I couldn't afford it at the time."
(stays focused on that time)