Q:

Why or when does battery capacity matter in a series circuit? 

I have 3 sets of doubled 18650 cells from a laptop battery, and I gather that it can be dangerous to use cells of different capacities in series.  This question is more about the theory than practice.  
I have tested each set of cells at 1amp continuous draw and as usual there is a minor differences in capacity; one lasts for 3h where as the other two last closer to 2h and 50m, if my math is correct that is 3000mah vs approximately 2860mah?  
These functioned in the laptop without issue and I'm assuming over time cell capacity diminishes and not exactly at the same rate per cell, this is obviously not dangerous, what is the difference between the results from these factory matched cells and unmatched or differently used ones that makes this safe?  
If I were to put these cells into a different laptop battery harness with their difference of capacity how would it differ from the original?  

A:

Lithium cells like 18650s are damaged if discharged too deeply. 
When you have several cells in series to make a battery, then you need to make sure that you monitor the voltage of every cell, and stop before the lowest goes below the minimum voltage, rather than the voltage of the whole battery. 
For instance, if you set an endpoint of 3v per cell, you might then look for 9v on your 3S battery.  However if there is a low capacity cell, the voltages might be 3.5v, 3.5v and 2v at a 9v total, and that cell is now permanently damaged. 
On charging, the situation is similar, no cell must go above 4.2v.  While a balance charger will try to maintain this, some balance chargers would only cope with or tolerate a nominal variation in capacities, and give up on a large variation in capacities, deeming it to indicate a damaged battery pack.