Forget...

They both read okay to me. 'about' seems extraneous when you re-write the sentence without it. To me, it's more a style issue than a grammar one.

It seems to be a difference in context, but only in certain circumstances. In your examples, it seems to mean the same thing. But check out these:

"I was so preoccupied, I forgot all about the party." Is a bit different, more emphatic, than "I was so preoccupied I forgot the party."

"She forgot his phone number as soon as he said it." Is quite a bit different than "I forgot about his phone number as soon as he said it." The second implies some sort of distraction instead of merely a lapse in memory.

Hope this makes sense.

I agree with CaroGirl in that there seems to be a basic, subtle difference when adding "about" and leaving it out. But in your examples: Forget the girl. Forget about the girl. It seems to work fine either way.

In the first, it seems to suggest you're to forget completely about the girl in question. In the second, it seems more that you're to forget the thoughts you were having regarding the girl.

??? it's just a gut reaction/guess on my part.

Ray, your examples use a slangy, figurative sense of "forget." "Forget the girl" and "forget about the girl" both mean "Never mind the girl; don't concern yourself with her." In literal use, when they refer to memory rather than attention, "forget" and "forget about" are different.

The notebook sentences might have various meanings, depending on context. There are some things that either sentence can mean and some things that only one of them can mean.

"I forgot the notebook": I left the notebook behind when I packed my briefcase, or I came home from the stationery store without the notebook I'd intended to buy, or I had no memory at Time B of a notebook I'd seen at Time A.

"I forgot about the notebook": I didn't think of the notebook at Time B, although I'd had it in mind at Time A (and that was why I didn't pack it or I came home from the store without one), or I failed to recall that this notebook was relevant to something I was considering ("Oh, that's right! We can find out the results of Professor X's experiment by checking his lab notebook"), or I had no memory at Time B of a notebook I'd seen at Time A because I hadn't thought about it during the interval.