I would say "after a while" and the context is short term... like "She gave up after a while." I would use "for the first time in a long time" in a different context as in giving something up and picking it up again over a span of months/years. You often hear it being used in a melancholic sense.

After a long time = a very long pause. He stopped. After a long time he started again. But we rarely use that phrase. We might say: He stopped. After a long pause he started again. I ran into a friend yesterday and we had coffee. That was the first time in a long time that we had a decent chat. （久しぶり）

Even though I'm not correct, I sometimes use the two interchangeably :/. I think that for the first phrase, there can be a possibility that you may have seen something/someone once or twice in a long time. However, for the second phrase, this is the very first time you are seeing something in a long time.