What is the difference between verbal language and non-verbal communication?

Both forms of communication transmit meanings. Both forms of communication include components which are learned. Language is learned, and people in difficult cultures may learn culture-specific gestures, like shrugging to communicate "I don't know." Language is a quintessentially symbolic communication system, but nonverbal communication isn't entirely lacking in symbols. For instance, in my culture, shaking the head from side to side is an arbitrary gesture that stands for the concept "no." That's symbolic communication.

There are many distinctions between these two forms of communication. Here are some, suggested by the work of Charles Hockett.

Structure

Language has structure—all languages have a syntax or grammar; rules of how to put together the units of language into a statement or utterance. Nonverbal communication has no such rules for stringing together units of meaning.

Displacement

With language, we can talk about things that aren't tangible, aren't present in the here and now, and might not even exist. Nonverbal communication is highly restricted. When you come home from school or work, you might be able to communicate that you are sad through nonverbal gestures, but you most likely can't communicate what events earlier that day are responsible for your sadness unless you use sign language. There are nonhuman communication systems that include limited displacement. Specifically, bees can communicate to each other about the location of food by performing a symbolic dance. I think it's difficult to find examples in human nonverbal communication (as opposed to examples that involve human sign language and other types of language that do not employ spoken words).

Productivity

With language, we can combine words to express new meanings. We can say things that nobody has ever said before, like "Gerbils are not willing to sit still and receive instruction for learning to read." Nonverbal communication is extremely limited in this respect. You might be able to communicate something new insofar as, say, nobody before has ever communicated the idea that your Aunt Mildred stinks, but language permits you to say things about Aunt Mildred that may never before have been expressed about any person.

Reflexiveness

You can use language to talk about language (as we are doing here). You can't use nonverbal communication to talk about nonverbal communication.