"Come in" is permission, offered to someone who has asked for it (by knocking, for example). Unsolicited, it sounds imperative, or presumptuous; though of course this can be moderated by tone of voice or other context.
"Come on in" is an invitation, much better as an unsolicited offer to someone who may not have been intending to enter. Offered to someone who has presented themselves to your door, it rather conveys that you'd welcome them even if they hadn't asked to do so. And it suggests that they will be joining you in a shared experience, rather than just entering a space for which you are the gatekeeper.
"Come on" :used for telling someone you do not believe what the person is saying or you think the person is not being serious: Oh, come on
or to encourage them to do something they do not want to do. Come on Doreen, let's dance. You say `Come on' to someone to encourage them to hurry up. Come on, darling, we'll be late.
