What are the advantages and disadvantages of these two methods to store your coins?
Hardware we have full control to our coin, maybe pros that we save coins in exchange is we can sell it immediately if needed. Because if save in our own wallet, that means more fees that will be used for transaction especially if we are trade and must doing deposit and withdrawal coins.
If that is so just because of the fess, I won't hesitate to spend less $1-2 for the fess rather than your whole crypto earning will disappear once the exchangers are becomes compromised. It's been always good to store your coin in a hardware wallet or any wallet that you have full control over your private key, in that way your coins are safe.
It could be the advantage of an exchange is the quick trade into fiat if you need it but it's just a matter of an hour if you'll wait while using hardware wallets. The most important factor is the safeguard of your portfolio and avoiding possible scams or even exchange becomes compromised and your coin might be stuck.