I don't see a great deal of difference between a request and an invitation -- none that could be communicated entirely by a modal verb, at least. The difference seems to lie in what you expect the addressee to do.
Request:
[Can/Could] you please come to my party tonight? I need someone to help me set up the sound system.
Invitation:
I'm having a party tonight, and [I'd / I would] like (to invite) you to come. I hope you can [come / make it].
CJ