If specification is 700mAH batteries can you use 850mAH batteries?

Sizing BatteriesIF there is room [physically] for an 850mAH battery to fit instead of a 700mAH battery, and it is the same type of battery [Lead-Acid, Ni-Cad, Gel Cell, etc.] then yes you can make the substitution. The 850mAH battery is just electrically slightly larger than the 700mAH battery.

The numerical designations for the two battery examples you give in your question mean that the first battery will deliver 700 milliamps [700/1000th of an Amp] for a period of one hour. The same is true for the second battery except that it will delivery slightly more current[850mA] for one hour, or the same 700mAH rate for a slightly longer time.