Whoa! Hold on. Is there a difference (when talking about keys and addresses), between Bitcoin and other coins?
What I've seen is that, to transfer your coins from your exchange wallet to your software wallet, you use your software wallet's public key, and to transfer funds from your software wallet to your exchange's wallet, you use the exchange wallet's public key. Basically meaning you can keep your funds safe by never using your software wallet's private key. Do you need both?
A cryptocurrency wallet is software that manages your private keys, signs the transactions you send to an exchange, generates the addresses you give an exchange to receive coins. Every time you send a transaction from your wallet, a private key is used to sign this transaction. Please note that you almost never see or interact with private keys directly, but your wallet does it every time you want to send coins to someone. An exchange does exactly the same: it signs a transaction with their private key and sends it to the address you gave them.