reference: https://www.youtube.com/watch?v=kE51N84hBxU
To check a block for integrity the nonce and the content are hashed together and then it is check if the numbers of zeros do match the number of zeros of the hash of the "guy" who mined the block.
Why do we not just compare the hashes after the mining, why do we only compare the number of zeros? I mean that does make little to no sense to me since comparing the complete hash does provide more security is no little to no much more effort.
The number of leading zeros is how Proof of Work works. Assume that it does not matter how many zeros the hash has. Then it would be trivial to find a block, as you could just use ANY nonce and provide the block (and some resulting hash).
However, if the requirement is "the hash has to start with 10 zeros", you have to try a lot of nonces (and do a lot of hashing) before you get a hash that matches the requirement.
Every participant checks if the hash has the required number of leading zeros.