Back and again have similar meanings, but there are some differences.

Back with a verb

Back is an adverb particle. When used with a verb, back suggests a return to an earlier situation. It may also suggest a movement in the opposite direction to an earlier movement or similar ideas.

Put the book back on the shelf once you have finished with it.
If he hits you, don’t hit him back. (= Don’t hit him in return.)
Give me my watch back.

Again with a verb

When used with a verb, again usually suggests repetition.

I did not hear what you said, please say it again. (NOT …please say it back.)
That was a lovely song. Can you play it again?
He was really bad-tempered. I am not going to invite him again.

Sell back and sell again

When you sell something back, you sell it to the same person.

Cases when back is not possible

Back is not used when the verb itself suggests the idea of ‘return to an earlier situation’. For example, back is not used with the verb return.

With adverb particles

Both back and again can be used with adverb particles like down, out, to and in with similar meanings.

He stood up and then sat back down. = He stood up and then sat down again.
I went back to sleep. = I went to sleep again.