They can mean the same. But a body part being 'asleep' is a special kind of numb caused by specific behavior. You can always 'wake up' a sleepy hand, but not so with a 'numb hand'. For example, a pain relief injection would cause numbness, but never cause you foot to go asleep.
“numb” typically comes from the cold. When they are numb, you can slap your feet or poke it with a needle and feel nothing. You feel numb in your mouth if you have to get a tooth pulled because of the stuff they inject into your gums. You do not feel a thing.
“asleep” in this case happens when your feet lose blood flow and get tingly. It happens often with your arms when you sleep on it funny and wake up and you have a hard time moving it and/or it’s all tingly and sometimes feels like a thousand tiny needles are poking you. If you touch your arm/foot when it’s asleep, you will feet a radiating tingling sensation. And over the next minute or two you will get the blood flowing through that limb and the tingliness will go away.