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... while his car was slightly faster than mine, it slowed down significantly within 3 5-lap races, while my much heavier car ran great for a good 10 races before slowing down.
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Clearly what is happening here is that, while he has batteries that normally would give him longer life, he is using MORE CURRENT than you; which means his battery life ends up much less. As Silentbob343 quotes from HowStuffWorks.com, depending on the maximum current rating for those batteries, the 850 mAh your buddy is using can be sliced many different ways; including sacrificing running time for more current.
And as Silentbob343 tried to explain, it is not the THEORETICAL current that determines what wattage the motor can use, it is the AVAILABLE current that determines what wattage the motor can use. If the maximum possible current a battery can give at once is 0.2 mA, it doesn’t matter how many mAh the battery has; it is bad for racing.
Check what the maximum current rating (or maximum wattage and work backwards) of his batteries is, find out whether he has FETs installed, and find out what kind of motor windings he is using. He is using more current somehow; find out why.
Silentbob343, can the stock FETs really handle 7 amps? And when do you have to start using FET upgrades?