This one's more of a fair comparison imho. Both are very similar in lots of ways. Given a good condition example of each, I can't say that I see much difference- both are excellent pumpguns from a bygone era.
Both have steel receivers. Both use the same sort of tipping bolt locking system. Both have a single action bar. A major difference is that the High Standard design has a side ejection port and loads from the bottom, while the 37 both loads and ejects from the bottom, making it a bit more friendly for the left handed shooter.
The High Standard design (prototyped in 1950, by Fred Humeston, formerly of Winchester- his name is on the basic patent covering the M1 carbine) is a little later than the 37 (1937), but both are from the same basic era of American manufacturing.
Another difference is that the 37 is a takedown gun- its barrel can be easily removed. The High Standard has a screw-in barrel that is more or less permanently affixed, save for a handful of late production Deluxe models.
IMHO a lot more people are familiar with the 37, so I will spend a bit of time discussing the High Standard. I've been working with one of the High Standard riot guns lately, just for experiment's sake. Here at hand in the position usually occupied by one of the 870 house guns is a recently acquired High Standard Flite King Riot Model K-102, a retired veteran of the Cumberland County (NC) Sheriff's Department (so my dealer told me). It has a 20" bead sighted CYL bore barrel, a six shot magazine, 14 1/4" LOP stock, and its brown plastic buttplate is mounted on plain walnut furniture. Safety is a crossbolt mounted on the front of the trigger guard, action release is on the left at the rear of the trigger guard. This one is parkerized, I don't think that finish was original on the High Standards because every other one- which is not many- I have seen was blued. I don't know, I'm no High Standard expert.
High Standard started producing factory original riot guns in 1962. Production of the Flite King ended in 1976.
Far as I can tell the two are of equal quality in design and production, some favor will accrue to the 37 from those who love the design or are southpaws.
FWIW,
lpl
http://www.histandard.info/models/shotguns/pump.html