In Proof of Work (PoW) new blocks are generated by miners who solve cryptographic puzzles and include the solution in the block header. Peers only accept blocks with valid solutions. In the case of a fork peers choose the longest chain.
In Proof of Stake (PoS) new blocks are generated by validators who stake their coins on the chain. In the case of a fork peers choose the longest chain.
The problem is that with PoS, in the case of a fork there is no penalty for a validator who decides to just stake on all forks. This is the nothing-at-stake-problem.
Ethereum’s solution to the nothing-at-stake problem is the Casper protocol, basically if someone stakes the same coins on two different forks they lose all their coins on both forks. In principle this solves the problem.
But there is one (big) problem: the staking transaction can’t be fungible. Now not only you need a special kind of not-private transaction but you also need the most important participants in the network to forego their privacy. IMO PoS is fundamentally incompatible with privacy coins.
With PoW nothing else needs to be known about who mined the block. It could have been mined with any kind of machine, any kind of power source and anywhere in the world. It also doesn’t matter how much capital the miner invested in the rig and where that capital came from. Blocks are truly fungible.