Wallet's main purpose is to allow you to store your coins and use them, by receiving and emitting transactions.

Exchange's purpose is to allow you to trade crypto for fiat currency (or possibly other values), and vice versa. Most of them technically also are wallets.

The main disadvantage with exchange wallets is you never know what will happen to your coins if there is a hack or other. So if you have plans to trade the coin keep in exchange wallets or if you want to hold the coins in long term keep in wallets.

But if you are holding coins in the wallets you need to save all private keys, seeds etc. otherwise you loose all your coins. If you lost coins in an exchange then the exchange may be responsible for that.