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What is the different between "forget to do sth." and "forget doing sth."

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When the most usual meaning of "forget" is intended, the infinitive follows. This is by far the most typical usage.

I hope I don't forget to lock the door.
I forgot to take my wallet this morning.
Don't forget to say "Thank you" when you see her tomorrow.
Sam is always forgetting to take his pills.

The gerund is not as often used. It gives a slightly different meaning to "forget". In this construction, the idea is that there is no hope of doing something, that one may as well abandon the prospect of doing it. In this structure, "can" is usually used with "forget".

Gary is very angry about what happened last night. You can forget asking him to pay back the loan now. Give him a few days to cool down.
It's obvious it's going to rain. I guess we can forget painting the house today.

Emotion: geeked
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I've understood. Thank you very muchEmotion: smile