It seems they both mean never mind in the following examples:
Sorry for what happened yesterday
Can I buy you a drink?
Forget about it.
What are the differences between forget about it and forget it?
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For me, forget about it is used more or less neutrally to mean Nevermind, don't worry about it.For example, you are at a restaurant and you order a Coke with your meal. Your waitress brings your meal, along with the bill, but not the Coke. She realizes it immediately and says she'll go get it, but you say, "No no, forget about it! It's not important", because you don't want to trouble her with writing up a new bill just for a Coke.
Forget it, on the other hand, is used negatively, with a strong hint of exasperation/annoyance/anger. For example, you are at a restaurant and you get into an argument with the waitress because she charged you for a Coke that you never ordered or even received. In the end, you say, "Ok, forget it. I'll pay for the damn Coke."
In literal usage, the oblivion indicated by "forget about it" is perhaps of broader scope than the oblivion indicated by "forget it". Eg: forgetting about a number suggests that one is no longer aware even that one is supposed to remember it.
Whether that nice distinction is maintained in the figurative assurances "Forget it" and "Forget about it" (so that "forget about it" implies a more complete absolution) is doubtful.