There are some contexts where both words can be used to indicate something never happened again. For example, when announcing their intent to quit smoking, someone might say:
I'm not going to smoke anymore.
I'm not going to smoke again.
Both of those sentences work fine.
However, sometimes the words don't function as exact synonyms, and the sentence needs to be tweaked if you are going to use one instead of the other. For example, I might say:
Bob doesn't work here anymore.
In that sentence, I can't really substitute anymore with again. However, if Bob got fired, I suppose I could say:
Bob won't work here again.
And when dealing with something in the positive instead of the negative, anymore can't be used. For example, if I say:
Teresa sprained her ankle again.
I can't make a substitution and say:
Teresa sprained her ankle anymore.
In your particular example, if you're telling someone about your nightmare the next morning, either of these work fine:
I woke up abruptly from deep sleep last night and couldn’t get back to sleep again.
I woke up abruptly from deep sleep last night and couldn’t sleep anymore.
(For some reason, I prefer again when get back to is used, and anymore when it isn't, but that's just a personal preference. I can't think of a grammatical justification for it.)