'A lot of' is to be used in positive sentences such as: They invited a lot of people to the party.
'Many' is to be used in negative sentences and questions such as: They didn't invite many people do the party. Did you invite many people to the party?

My question to you all out there is; should I be correcting students who say, 'They invited many people to the party'?
I don't think I should but the course book begs to differ.

I asked for some help with a grammar qestion a while ago and I received great feedback, thankyou.