MyCelium is a wallet and coinbase is primarily an exchange. Coinbase can and will collect personal data about you, have certain level of control over your coins as well.
Coinbase also does not offer any sort of discounts for on chain transactions as well. MyCelium, though being less polished, still offers far more security as well as privacy.
There are two different apps - Coinbase (exchange) and Coinbase Wallet. The last one is Coinbase Wallet is a non-custodial product. So it doesn't keep any users data.