> Cardano I believe uses PoS of some sort.
Yes, but smart contract languages (Plutus in Cardano's case) generally are unrelated to the specific consensus algorithm of the blockchain platform.
> What is the thinking about what keeps these distributed application servers running? Is it still mainly the mining reward model?
In general, public blockchain platforms run by incentivizing the operators within the system itself. PoS and PoW are different versions of this (staking v mining).
> Are there applications the distributed model enables that the traditional does not?
Crypto, and DeFi (decentralized finance), is conducting a huge "live beta" in decentralized corporate governance and finance. There is a crapton of innovation happening with financial engineering and exchanges, completely new models like Uniswap (constant product DEXs) or new versions of existing things, like on-chain options.
Is it world-changing? No, or not yet certainly. But by construction, DeFi can only happen on a decentralized smart-contract blockchain, so it's genuinely new.