The white wire is (99.5 % certainly) a temperature measuring thermistor connection that helps your system not destroy the battery and in the process create smouldering ruin. If your battery gets too hot the system will stop charging - if it 'knows' the battery is too hot. If not ... .
Batteries without this 3rd connection are relatively rare and it may well be a sign that the maker is not overly worried about smouldering ruins after he has sold you the battery.
There is a chance that the new battery is "smart" and has an internal processor and signals on the battery leads BUT the chances of this being true are minimal.
Increasing capacity from 300 mAh to 3000 mAh will work OK if both are single cell (3.6V nominal) LiIon or LiPo as they probably are. The charger for the small battery will take about 10x as long to charge the new one (as 3000/300 = 10 :-) ) BUT will probably never stop charging as Vcharge does not fall to expected low level. If left on charge for long periods (say over 24 hours) it MAY damage the large battery.
Using a larger mA charger is a good idea.