Yup, the cells are wired in parallel so the only reason you'd want more cells is to multiply the available voltage. Resulting in what you have observed.
It's not quite that simple. The S stands for cells in series. Each lipo cell is nominally 3.7 volts. So 2S is 7.4V and 6S is 22.2V. There are no cells in parallel unless specified in the battery description.
Both are 3000mAh and therorteicall can deliver 3000mA or 3A for 1 hour.
If you draw 3A from the 2s you will consume 3 x 7.4 = 22 watts of power. If you do the same with the 6s you will draw 67 watts of power. Both will deliver that for the same amount of time because it's based on capacity. You can come up with scenarios where the 2S will draw more power for a shorter period of time than the 6S, it really depends on what you are powering.
That being said, higher S is typically used for higher power applications.
DamoRC
Edit. Oops - typing at the same time as Hai-Lee