What is the difference between hexanes and petroleum ether
Petroleum ether:
This is a mixture of a lot of different compounds (aliphatic hydrocarbons in this case) that as a mixture have a boiling point within a specified range. The contents are dealt with as having low amounts of aromatic compounds and high amounts of this and that, not one specific formula. It's like a blackbox to me and can be used for crystallisations that are usually easy to do.

Hexane:
In the synthesis laboratory I worked in, hexane was dealt with as we would deal with petroleum ether, since it was a mixture the many possible constitutional isomers and not pure n-hexane. We bought the technical grade hexane in our chemical store and purified it with a solvent purification system to have it dry and compatible for glove box use. Since they appeared so clean in our Schlenk flasks I thought in the beginning that it was pure n-hexane but a simple 1H NMR spectrum taught me otherwise.
The hexanes mixture was used for medium difficulty crystallisations.

Pentane:
The pentane we bought and purified, however, was n-pentane and was used for difficult crystallisations that were not easy to reproduce. Already among organometallic compounds there are examples in which obtaining crystals at all is an enormous struggle and having single crystal quality is simply a matter of hope.

Therefore, by going from petroleum ether to hexane to n-hexane (or n-pentane) is a matter of going from a wide mixture to a compound to a pure chemical.