They are similar. Not exactly the same, but quite similar. The concepts are roughly the same, as they both describe the dark part of an object furthest from a given light source. The usages of each of these words are different, however. For example:
When talking about the dark form that follows the feet of every person and animal, that is ALWAYS called a shadow: My shadow, your shadow, the dog's shadow, etc.
You would also say that everything "casts a shadow." When the verb "cast" is used, then the noun "shadow" is used. I don't think I have ever heard "cast shade."
"Shade" is most often used to describe the part of a building, tree or other large object that people use to shield themselves from the son's light and heat. For example, one might say:
"Let's go have our picnic over there in the shade." (perfectly fine. most people would say it just like this)
"Let's go have our picnic over there in the shadow." (This would probably never be said. It sounds strange to me, at least.)
Both "shade" and "shadow" are used in numerous different contexts. However, in general terms, think of it like this.
A "shadow" is normally used when talking about darkness in relationship to a particular person, animal or thing.
"Shade" is normally used when talking about a general location, where there is a darkness. (The objects that cause said darkness are not important)
I hope this helps. If you have any questions, please let me know.