Actually, most of the “standing” ship’s officers in that era weren’t “commissioned” in the sense of given a commission by the Queen to fight Her Majesty’s enemies, etc… but rather hired through a warrant.
As I understand it, the main difference is that the Admiralty would issue a warrant for the hiring of surgeons, gunners, pursers, sailing masters, and so on, and captains would hire them under the authority of the warrant. Meanwhile, officers were directly commissioned by the Queen, with different authority and command responsibility deriving from that. Warrant officers weren’t in the chain of command as such- they answered to the captain of the ship, whoever that may end up being.
Historically warrants were used to hire all sorts of specialists and slot them into military roles- it was a holdover from the days when you had the “Ship’s company” commanded by a Captain, and he hired the specialists needed to make the ship sail and fight, as he was a gentleman, not a sailor or whatever else.
These days, the difference is a little bit more academic, in that warrant officers have a separate commission- the ranks are effectively for highly specialized positions that are not necessarily directly in the chain of command, but require respect and treatment like officers - specialists like helicopter pilots in the Army, or Infantry Weapons Officers in the USMC, for example.
I still think that Starfleet would probably have looked askance at the “temporary Ensign” appointment of Wesley, unless they were so far outside Starfleet’s reach and influence that Picard needed to appoint an Ensign to fill a role for some reason, and there wasn’t a qualified petty officer or midshipman or something who could fill the role. And I never got the impression that the Enterprise was out there so far that they were ever in the position where they couldn’t just appoint a petty officer to be helmsman for a few months until they could pick up a new one at the nearest Starbase or passing Starfleet ship or whatever. It seemed mostly like a way to keep Wesley in the main cast and on the bridge, not anything actually realistic, even by Star Trek standards.