Mining is voting weighted by wasted CPU power. Stake is voting weighted by wealth. That's all, really.

Both are used to build decentralized networks, since they allow it to pick a random participant to order the previous X transactions (aka block), without being susceptible to sybil attacks. PoS is much less energy-hungry than PoW, but it has a problem: how do you distribute money fairly, to begin with? Which is why Ethereum started with a few years of PoW to only then shift to PoS. Pretty clever IMO.