#clells x capacity is constant
For example 5 (cells) x 2800 (mAh)= 14000 6(cells) x 2333 (mAh)=14000 These two battery packs can be considered equivalent and store the same amount of power.
If you need nose weight it is best to put a bigger motor. First the weight is placed where it makes the most of it, second it will run cool which means no lost energy which will give more battery head room, third the ESC will work easier.
Another thing to consider going from 5S to 6S is the motor Kv. I like to run the motors slower with props in the 6-7 pitch range. In general I think 11000rpm as my base to calculate the Kv.
For example 5S=18.5V 11000/18.5=595Kv so any motor above 590Kv to low 600Kv will be fine on 5S
Similar 6S=22.2V 11000/22.2=495Kv so any motor above 500Kv to low 500Kv will be fine on 6S
As for effect think in terms of power demand from the system. Regardless of what you use in flight at the same moment and place in flight there is some power requirement to get the job done, this power is the product of Voltage and Amps, so if voltage goes up then amps will come down. So consequently if you rum higher voltage the amps will come down proportionally which mean the battery will have an easier workload and will be less hot. Keep in mind that during a flight there is some power requirement to complete the flight, if you have heat in the system (lost energy) that will be tagged in addition to the power required for the flight, simply put the battery will have to be bigger heavier and more expensive.
The easiest best way to save weight is to oversize the motor. If you do this the ESC Amp requirement will drop (lower weight this alone perhaps will compensate for motor weight increase) with lower Amp requirement means the C rating of the battery can go down (even if you end up with the same capacity and cell count a lower C rated battery is lighter), and with less power required for the motor to get the job done that means the battery will require less capacity (that is cheaper by far).
I hope this clarifies some things and help.