Proof of Work has many interesting properties. Bitcoin uses a specific PoW puzzle based on SHA-256 hash function in order enforce the miner's commitment of hashing power solely to the security of the bitcoin blockchain; also PoW is more resilient in the context of some attack vectors such as Sybil attacks; it also has some downsides of course, but some of them can be refuted from an economical perspective - bitcoin is backed by raw processing power.
Some variations of the PoS model that do not have inherently mathematical proofs and usually are inspired from real world scenarios.