A canoe is paddled with a one sided paddle, usually while sitting on either a raised bench or one or both of your knees. A kayak is paddled with a two sided paddle, usually while sitting on a seat on the floor of the boat. They're approaches to using different muscle groups. The main power of a kayaker comes out of rotation of the upper body, while in canoeing there's more of a front to back movement involved.

That's it. If you take a freestyle kayak, take out the seat and put in some foam knee supports instead you've converted it to a canoe, you're now eligible to compete in the c1 class instead of the k1 class. (The 1 is the number of people in the boat.) The hull shape and such are completely secondary.

If you have any specific boats to ask about, what their origins are or such, I'd be happy to take a look. But this is the difference.