If the 125 psi is measured at the tool while its running, then how it gets there doesn't matter.
I suspect the issue between your home and shop may be the air regulator capacity and the volume of air in the pipes and hoses that act like a buffer tank. Nothing else should be much of a factor, not the compressor size, tank size, or even the size of the air lines in terms of flow rates (the size is adequate for the flow required by a 3/8 impact).
I haven't been having any issues with my setup, other than the shop air being a bit stronger.
Your tool wouldn't have 125psi while running if your lines, valves and such aren't capable of keeping up with the flow the tool requires. Larger tanks typically have larger valves and hard lines on them.
If you want to test it out for yourself, go buy this HF air line
http://www.harborfreight.com/1-4-quarter-inch-x-25-ft-self-coiling-air-hose-47.html
and test the torque of your impact, or the speed of your air ratchet.
Then replace the line, and nothing else with this:
http://www.harborfreight.com/3-8-eighth-inch-x-25-ft-reinforced-rubber-air-hose-42184.html
I've already done this. (the 3/8" was actually a goodyear, but that shouldn't matter) Tool output was noticeably stronger with just this change.
An old schooler that installed another one of my shop's air system explained all this to me once. PSI does make a difference, but so does the volume the lines/valve/hose can flow (including the ones on the tank). He said that the cfm or volume of air that flows is actually more important, but that's obviously a matter of opinion, although I share his opinion after seeing it first hand.
He went on to explain that this is why my air tools seemed different from one shop to the next even though they all used identical air pressure. The lines weren't all the same length, or even diameter, and even though you could plug a guage in that showed the lines pressurized with 125 in either place, they couldn't keep that 125 up with a tool with high air consumption.
Take what i'm saying at face value, or don't. If you test for yourself, I think you'll be surprised at the results though...