These terms are widely used for many and varied concepts. They are roughly synonymous and often used interchangeably.
In some contexts, a channel refers to a subdivision of a large communication medium. For instance, radio and TV stations use "channels" to describe the frequency-division multiplexing approach to separating signals. The AMQP message-bus protocol uses channels to multiplex traffic over a TCP session.
In computer science, the term channel sometimes refers to a message-oriented pipe between communicating end-points. Tony Hoare's CSP (from which Occam, Limbo, and Google's Go are derived) use channels as the fundamental unit of communication and synchronisation.
The term stream, OTOH, tends to refer more to a byte-oriented communication pipe, such as a TCP socket, which delivers a continuous stream of bytes or characters, without a clear subdivision that separates one message from another.