The biggest advantage of an exchange is that it is user-friendly and much better if you're new to crypto. There's no doubt. Ask anyone that's never owned or dealt with crypto before to download Metamask and they'll be like 'what the heck is that?', ask them to download Coinbase and they'll be like, "okay, it seems easy enough."
Another major advantage is that exchanges generally allow you to convert between coins and tokens that operate on different blockchains, that's because these transactions happen off-chain. Imagine having to transfer ETH from Exchange A to Exchange B, it'd cost so much in fees and it'd be relatively slow. Whereas if you convert that ETH into, say, Algorand, (provided both Exchanges support Algorand of course) before you send it the transaction will be instantaneous, and basically free, and 30 seconds later you can convert it back to ETH on the new exchange you send funds to. The third advantage I can think of is that you can easily top up and withdraw funds from your account. You can do that with wallets as well, but it's usually a bit more complicated.
The main issue with exchanges is that it is, technically speaking, true: you don't own your coins. The exchange can be hacked, it can go bankrupt, etc. Obviously, this is a much bigger risk if you're dealing with a dodgy exchange, whereas major exchanges such as Coinbase or Binance are usually well preotected and provided you use a complex password, 2FA, etc, your funds should be safe. But it is true, they're not technically yours, and they're not theoretically 100% safe.
The second most important disadvantage I can think of is that exchanges do not, generally speaking, provide true DeFi services. Most exchanges provide some form of 'Earn' products and some even allow you to stake, but it's not 100% decentralised.
The biggest advantage with wallets, by far, is security. If you can learn your seedphrase by heart, it is virtually impossible for anyone to gain access to your wallet. There's no way anyone can ever do that, short of forcing you to give up your keys (ie, by force), it just can't be done.
Another major advantage is that with wallets your financial experience is truly decentralised, allowing you full access, and control, over your funds and the things you can do with them.
I can think of two major drawbacks. One, and this is quite obvious, if you lose your private keys and seedphrase, no one can help you. It's important to understand this because I know that most people on this website already know this, but newcomers generally don't, and it takes some time to process the information. And then two, should you choose to cash out and withdraw your funds, for any reason, there's no doubt that is much easier to achieve with exchanges, some even offer debit cards, whereas you can't really do that with wallets.
Do you normally use wallets or exchanges? I use both. If there's anything you wanna add to the list, do let me know in the comments. Thank you.
I've crossposted this article on my blog bitcoinea and LeoFinance