A canoe is a boat designed to be best paddled with a canoe paddle, a kayak is a boat designed to be best paddled with a kayak paddle. You can propel a kayak with a canoe paddle, or vice versa, or fit either one with a sail, or a motor, or oars, or a pole, or throw a sea anchor as far as you can and reel it in—but as far as putting the name on the boat I’d say it’s which propulsion method (canoe style or kayak style) is the one for which the boat has been best designed.
The paddle and associated body position have implications for the boat design and give rise to the most visible distinguishing characteristics, principally the height of the sides of the boat and (closely related) whether the deck is covered or not, but also how slender / tapered the hull can be and still be stable.
A canoe paddler sits or kneels with their body’s torso relatively high and holds the paddle close to vertical. A kayak paddler sits low with a paddle at a much lower angle to the water (the exact angle may vary between long distance touring and racing/other extreme performance situations). What that posture means is a canoe can have high sides without interfering with the paddle stroke. What high sides means is that the craft can handle larger waves before it needs to be covered to keep water out. A kayak, with its low sides, is typically covered by a deck to keep the water out, and the paddler sits inside through a hole in the deck, often further sealed by a spray skirt, or in a well in the fully enclosed deck (“sit on top kayak”).
The paddler height has implications for the stability of the boat, and thus a kayak with its low-seated paddler can typically have much less width (either narrower at its widest point, or with pronounced “hips” and a fine bow and stern rather than being wide along most of its length).
Additionally, the two-bladed paddle makes it much easier to make instantaneous, intuitive “brace” strokes on either side of the boat to stay upright if tipped, making even further instability practical for most paddlers (achieving this with a canoe paddle—rapidly bringing the paddle across the boat as needed—requires more skill and dexterity). It’s also possible for a skilled paddler to paddle an even more narrow hull that cuts even more efficiently through wind and surf.
Low seat height, enclosed deck and ability to brace stroke on either side instantly make kayaks the more common choice for extreme whitewater although certainly whitewater canoes (typically fitted with flotation) do exist.
Lack of a fixed deck, tall sides and width that is maintained for much of the length of the hull make canoes the more common choice for conveniently hauling lots of heavy, bulky gear (/harvested game)
However, another side effect of the kayak paddle is many people find it a much shorter learning curve compared to a canoe in terms of learning to control the direction of the boat, especially as the seated position allows the feet to be used with a pedal-operated rudder. This has given rise to an enormous number of recreational and fishing kayaks, with wide hulls which are much more stable, although lacking some of the advantages of traditional kayak hulls for cutting through wind and waves.