A PAW is an animal's foot: the fleshy, bony part, often with pads underneath (as they don't wear shoes!). It's not the whole limb/leg - just the 'foot' part. When an animal walks in soft soil or sand it leaves pawprints.
A claw is a nail - formed into a long, often curved shape. Claws can be very sharp. They can scratch you and draw blood.
Typically, an animal that has paws has claws too.
Bird's don't have paws. They have feet. But as a bird's foot is very thin and bony, usually with large claws, it may seem as if a bird's foot is all claws and not much else! You don't call a bird's foot a paw. Much too bony!
Elephants don't have paws, they just have feet. Horses have hooves. For it to be called a 'paw' it needs to have soft parts (pads) underneath, like in dogs and cats, and claws attached.
Paw and claw can be used as verbs. If a dog paws me, I don't mind. If a dog claws me, it hurts.