I have come across this rather more times than I’d like recently. If anyone remembers the short-lived Libro character, Ranting Ron, who turned out to be too negative for this nice, friendly, supportive blog, he would have been ranting about this one ..
Invite is a verb. It means to ask someone to go somewhere or do something, and can be informal (“she invited me to go for a coffee after yoga”) or formal (“Mr and Mrs Perkins invite you to the wedding of their daughter, Polly”). It can also be used to refer to eliciting a response of some kind – “I would like to invite questions from the audience”. Full stop.
Invitation is a noun. It’s a written or verbal request inviting someone to go somewhere or do something. It’s the bit of card that is used to do this, which Mr and Mrs Perkins will send to you when they want to invite you to their daughter’s wedding. Invitation is the noun to invite’s verb.
Now, I have to admit that the esteemed Oxford dictionaries do admit that there is an Informal use of invite to mean invitation. But I don’t think we need that, do we? Let’s keep the distinction and keep it all nice. OK?