Where your crypto actually lives
Crypto doesn't live inside an app. It's recorded on the network (the blockchain) and tied to an address. Whoever holds that address's private key is the one who can move the balance.
A wallet, then, is what stores that key and signs your transactions. The question that matters is: who holds the key?
Exchange: someone holds it for you
On an exchange, the company holds the key. You see a balance in the app, but the exchange is the one actually moving the coins. It's convenient (you can reset your password, buy with a bank transfer), but your balance depends on the company: it can freeze your account, ask for documents or, in extreme cases, go bankrupt.
Self-custody: the key is yours
In a self-custody wallet, the key is created on your device and never leaves it. Nobody else can move, block or freeze your balance, not even the company that built the app.
The trade-off is that the responsibility is yours: if you lose the key (or the recovery phrase that represents it), there's no "forgot my password". Nobody can recover it for you.
Which one should you choose?
Many people use both: an exchange to buy and sell with their local currency, and a self-custody wallet to store crypto and to pay and get paid by other people.
- Exchange: buy and sell with a bank transfer, account recovery, but your balance stays with the company.
- Self-custody: full control and no middleman, but you keep the recovery phrase safe.