There was an article about this in RCCA a year or so ago about higher capacity batteries having more "oomph". This all applies to batteries powering RC cars, not servos. Take the C rating (20c, 40c, etc) and multiply by the battery capacity (5000mah, 5500, mah,etc) and you will get the max discharge rate in amps. So....
1) 5000mah 20c battery can discharge at 100 amps safely.
2) 5000mah 40c battery can discharge at 200 amps safely.
Now lets take two different capacity batteries with the same C rating:
1) 3000mah 20c battery can discharge at 60 amps safely.
2) 5000mah 20c battery can discharge at 100 amps safely.
So the same C rating doesn't mean they can accelerate a vehicle under load the same. The higher capacity battery will win everytime.
Now this doesn't apply to receiver batteries at all. We don't have that kind of load on the battery. So a 6v 2000mah battery will perform the same as a 6v 5000 mah battery.