TCP vs UDP - What is a TCP connection?

What exactly is a TCP connection? I understand there isn't a physical connection from the client to server. Is this connection just the client's socket being linked with the new socket created by the server after the three-way-handshake? Thereafter once the "connection" is set up, the sockets on either ends of the connection then know where to send their packets.

How does this differ from the way UDP functions other than the initial handshake with TCP? Is it that each server socket only has one client that sends packets to that particular socket?

What are some possible advantages of having a dedicated connection between hosts? My understanding of TCP and UDP is still very basic, so broad generalizations should suffice.