The M37 had two production runs... 1951-1956 and 1958-1968 (if my memory serves...). Bassically they are the same truck with a couple of minor differences that, considering depot rebuilds/MWO applications/modifications by various owners, make them a confused jumble similar to most other M series trucks on the market...
In short, the M37 had a transmission that was only identified by the New Process casting number 88950, the M37B1 had a NP420. Only differences in the transmissions are that the bearing retainer nut on the input shaft is left thread on one, right thread on the other; fill plug changes side from early to late, angle of gear that drives the PTO changed about a degree, longer shift lever on the later model, different internal shift fork arangment (three decending fingers on the NP420).
The M37 had the spare mounted in the bed with the troop seat on the passenger side shortened and a "jump seat" installed that could be raised on combined with the troop seat to make the troop seat full length when the tire was not mounted, M37B1 had the tire mounted over the driver door using the same hardware as the ambulance (and some of the specialty bodied trucks).
There is some indication that the steel compostion (alloy) was different, some folks claim that the later models don't rust as much (I disagree, but will not discuss steel alloy composition and properties here and now...).
Running production changes caused the vehicles to migrate from metal wiring harness connectors to rubber shells, instrument panel lighting changed, service brake lights got added tot the right side and turn signals got added, minor changes got made to the brake wheel cylinders, firewall mounted gas filter got deleted, etc.. None of these are model specific (and some such as the very early wheel cylinders and two plug three lever light switch are extremely short lived examples and very likely got changed on everything before long so unless you have a truck that got discharged very early, you likely won't see them...)
You really only need to worry about the M37 or M37B1 bit when you are checking the title against the dataplate, and when you are talking to somebody who knows (like when ordering parts...). Most everyone I know just calls them a M37 unless they call them "the little truck", "3/4 ton", or on bad days...."your jeep" or "nice Humvee" ...
Enjoy