do mAh on batteries (for home-use - nerf) make a difference?
More mAh means longer lasting

mAh means milliamp-hours, and is a measure of capacity. Milliamps are a measure of current, i.e. the amount of electricity flowing out of the battery, whereas Volts are a measure of the energy in the electricity. Milliamp-hours are a measure of how long the battery can provide a certain current.  A 1000mAh battery can provide 1000mA for 1 hour, or 500mA for 2 hours, or 3000mA for 20 minutes.

Caveats:


It's not really quite that simple with a real battery. If the battery is being discharged slowly, then probably you'll get the rated number of mAh.  If it's being discharged quickly, then you'll get less.  If it's being discharged very, very quickly you'll get a lot less, and maybe a fire. Look at "C" ratings for a measure of how quickly you can discharge a battery.
If you replace the 3.6V source the nerf gun was designed for with a 7.2V source, that will probably affect the current it takes.  So a 1000mAh 3.6V battery pack will last longer than a 1000mAh 7.2V pack.
Be wary of stated capacities.  It's not easy to measure, so it's something dodgy retailers lie about.  Batteries bought on well-known auction sites may have a much smaller capacity than advertised.