"Mining" is fundamentally the process of validating new blocks on the chain. Minting and distribution of new coins is a reward for mining on BTC. The miners aren't really "discovering" new bitcoins. The protocol mints them at a determined rate as new blocks are hashed and then the new coins are awarded to miners. The rate at which new BTC is minted steadily decreases to an eventual cap of 21 million coins. But the mining pool is also paid a per byte fee for each transaction in addition to receiving portions of newly minted coins for their work.
No mining = no new blocks = dead blockchain. The fees are how BTC will keep going once the last Bitcoin is minted. The main reason that Bitcoin mining isn't profitable for the little guy anymore is that there are massive commercial operations running millions of dollars worth of dedicated hardware. As a result, the little guy with a video card will almost never be the first to validate a block, so will almost never get rewarded for the work.
As you noted, Ethereum works slightly differently