A plug is either an object used to fill a hole, or a cord that plugs into a wall for electricity. Both these are nouns, but can be used a verb. Ex. "Can you please plug the computer in?" Connect means to join, unite or fasten together. People can connect with another, but you may also connect objects, like phones and internet. It is always a verb :) Ex. "I can really connect with you. We're very similar." "My internet is finally connected. I can talk to you on Skype."

Are you comparing the verbs? Plug can be both a verb and a noun. If comparing the verbs, then the difference is that plug is to shove something in to a hole or gap; and connect is to attach or join two or more things together. They are different actions.