Why would a miner mine with a difficulty other than 'the' difficulty? Why would a miner mine with a difficulty other than the network difficulty? Both of these questions describe miners mining in such a way. What's the benefit of this? What even does it mean?
To understand this we should first explain that a block with a high difficulty, let's say D, is also a share of low difficulty d &lt; D, therefore if we find such a block for difficulty D we automatically have a block of difficulty d. This also means that if I as a miner am looking for a share with difficulty d I might by chance find a block of difficulty D (with the probability d/D). This is exactly the reason why mining pools exist: they assign the miners the task to find blocks of difficulty d in the hope that they will eventually find one for D, i.e., the real network difficulty, which pays for the whole pool operation.

Commonly these blocks at a lower difficulty are called a share, and are used to gauge the hashing rate of the individual miner, giving the pool a way to determine who has what probability of finding the next block and distributing the reward accordingly.