If I remember right, Grade A has only a 25,000 PSI yield strength while Grade B has 35,000 PSI. But it's been a while since I read it, and I may be confusing grades/ types.
As JStephen pointed out, you could read the spec! Also available as SA53 in the ASME Boiler and Pressure Vessel Code Section IIA (material spec) or IID (SMYS, SMUT, allowable stresses). From IID, SA53A has SMYS = 30 ksi, SMUT = 48 ksi; SA53B has SMYS = 35 ksi, SMUT = 60 ksi.
Another significant difference is that I believe you'd have a hard time finding the A grade material. The B grade is much more common in my industry and probably costs less than the A grade simply due to availability.
Today, the lowest easily-avilable pipe is Grade B/X-42 dual stamped. I haven't seen a supply store carry straight Grade B in several years and I can't remember seeing Grade A in an Oil-Field supply store's yards for many years.
