How is ethereum mined?

So from my understanding, bitcoin is mined by PoW which is basically a mass brute force attack by miners to guess a random hash value, when this value is guessed, the block is validated and you are awarded BTC.

Now with ethereum, it is a proof of stake system rather than proof of work. When you are staked and selected as the transaction validator, what is the process used to validate this block? Are you still trying to guess hash values? If so why is this a better choice if you still have to waste energy to brute force the password (hash value) of these blocks?

Validator selection process aside, what is the difference between processing a block on a PoW system vs a PoS system?