"See you later" and "bye" may be redundant in meaning; however, they have some meaning if you examine the different levels of discourse in a conversation.
Here are two discourse-level observations. First, the sentences in conversations tend to get shorter to signal its end. (For example, think of a couple having a conversation in which one party answers in monosyllables. He may be indicating that he wants the conversation to end.)
Second, there are no gestures or body language over the telephone. The parties are signaling their mutual desire to end the conversation without recourse to the gestures of shifting weight, pointing feet toward the door, closing a notebook, or getting up.
"OK, talk to you next week."
"OK, See you."
Hope that helps. Bye.