For example was one company or even regiment or division issued with one or the other? I assume that some would pick up or trade different weapons. And new replacement troops arriving would have been issued differing things based when they came to the unit.
But was there atleast a effort to simplify spare parts needed in the unit quartermaster/armorer.
As far as I can tell, the Grease Gun and Thompson were not "segregated" because too few Grease Guns entered service to cause a significant problem; the 82nd and 101st Airborne Divisions even used a mix of Thompsons and Grease Guns when jumping into Normandy on D-Day, although the number of Thompsons was far greater than the number of Grease Guns.
About 660,000 Grease Guns were built during WWII, compared to over 1.5 million Thompsons. Even though nearly three quarters of a million Grease Guns were built, they suffered from serious production issues (read: Army bureaucracy) and were very slow to enter service. The M3A1 was not even approved for serial production until December 1944.
Remember that the M1/M1A1 Thompson was very expensive for a submachine gun (or a gun period), costing a staggering $209.00 in 1939. By 1942, the price went down to $70.00, and then to $45.00 by early 1944. In contrast, the M1 rifle cost $85.00 in 1942.
The M3/M3A1 Grease Gun, on the other hand, cost only about $20.00 to make. If a piece of it broke, it could simply be scrapped for other parts or thrown away and a new gun obtained.
At some point during the production of the M4 Sherman tank (possibly spring 1944), the specified crew personal weapon changed from the Thompson to the Grease Gun to reflect the Army's wish to gradually replace the Thompson. As before, submachine guns were mostly issued to drivers and armored vehicle crews.
Sources:
The US M3-3A1 Submachine Gun, by Frank Iannamico
Sherman: Design and Development, by Patrick Stansell and Kurt Laughlin
The M3 Submachine gun was actually a late war upgrade on the Thompson. They just weren't issued to most of the regular army before the war was over. They were mainly used by French Partisans and in the Pacific.
To explain why it's the better weapon...
The Allies saw the effectiveness of the German blowblack operated machine pistols and imitated the design with the 'Grease Gun.' It's much cheaper, smaller, and lighter. Plus, it could be easily converted to fire German 9x19mm rounds, so it'd be more effective for insurgents.
A buddy of mine owns an M3 and says that it's reasonably resistant to abuse, but it has occasional feeding issues related to bad magazines and is more likely to have a negligent discharge, and has a slower rate of fire than its predecessor.
Overall it's a decent weapon, the A1 series wasn't retired until the early 90's after Desert Storm.