Two big reasons. High fees when converting too and from fiat, and when you trade on the exchange you do not have positive ownership of the cryptos. You need to move them off the exchange into your own wallet first. Which you can do, but its a convoluted step for some people.
The positive side of things though is Coinbase is pretty easy to use, generally provides good information on various things, and for a casual investor (like myself), does not allow you to trade in more "experimental" stuff that might blow up in your face overnight. Its also got the imprimatur of institutional support as its a publicly listed company now. Meaning the SEC has at least glanced at their books. For the ANCAP crowd that counts a con, but for me its a pro. Some scratch and sniff crypto brokerages may offer little or no fees, but the trade off there is you can't truly trust them.