what is the difference in meaning of the sentences:
1. She forgot about meeting him.
2. She forgot to meet him.
There may be no difference. In most contexts there would not be a difference.
However, the expression "Forget about it" means to intentionally drop it, to dismiss it from your mind.
"How can I ever thank you enough? How can I repay you?"
"Ahh, fuggedaboud it. It was swell doin' a favor for a classy dame like you."
So it could perhaps be understood that "she forgot about meeting him" means she deliberately dismissed the idea of meeting him. To "forget ABOUT meeting him" is just off-kilter enough to ring that bell faintly in the background.
"Look, just forget about it. I'm not going to meet him and that's final."