I don't know this for sure the layout of Brinks style armored transports, but my first thought is that the Brinks trucks are designed to be secure from the outside. The fact that the MRAP (or any military style vehicle like a BearCat, etc...) often have turrets atop (with heavy caliber guns removed) that provide cover to an operator who may be providing long range cover to operators on the ground. I don't think Brinks style trucks offer this feature.
Doing some quick research, it looks like your standard Brinks truck only offers NIJ Level II/III armor. I would assume military vehicles that can withstand an IED will provide NIJ Level IV protection since the Lenco BearCat is also Level IV.
Also, the "buy in" for a Brinks truck is likely to be way more expensive than $0.00.
Lastly, I'm sure armored couriers put far more miles on their rigs than the military.