What are the advantages and disadvantages of these two methods to store your coins?
I was scrolling through the trading discussion section of the forum, and I came across a post that talked about an exchange that got hacked about recently. That is enough reason for you to avoid exchange wallet. You can only make use of exchange wallets when you want to trade your cryptocurrencies, but not for your main storage.
If you are using your exchange as your main storage, if that exchange should get hacked then everything you have worked for is gone. Avoid making use of exchanges as your main storage, try to make use of hardware wallets. Hardware wallets are more safer than making use of an exchange. Although I do know that every type of wallet can come with its own risk, but it’s worst when you are making use of an exchange.