In theory, 9 cell batteries have 50% more capacity than 6 cell batteries. Size? They are about 50% bigger as well. I have such for my work laptop. The larger batteries extend beyond the form of the laptop (the smaller batteries do not). This adds about 1 inch in front-to-back size to the system.
The number of cells in a battery isn't a good indicator of capacity. This is because not all cells have the same capacity.
The capacity of a laptop battery is usually measured in milli-amperes per hour (mAh). Knowing this value will help you to compare batteries.
For example, if a battery has 6 cells at 4800 mAh, it should last twice as long as one that is rated at 2400 mAh, or 33% longer than a 9 cell battery rated at 2400 mAh.
Adding more cells increases a battery's voltage. If you need to work out how many cells there are in a battery, divide the voltage by the nominal voltage of each cell. For lithium ion cells, I believe the nominal voltage is about 3.6 V.
Once you know how many cells a battery pack contains it shouldn't be too difficult to estimate the size of each cell.
Why wouldn't the cells have the same capacity? They're probably manufactured the same.
Also, I agree that the number of cells in a battery isn't the best indicator of capacity, but for laptops, it works because the voltage is constant. Additional cells are added in parallel and increase the capacity. You don't want extra cells to add voltage because the laptop has the same required input voltage for both battery packs. If you divide the voltage by the nominal voltage, you'll only get the number of cells that are in series.
The capacity in mAh is just the total energy stored I mentioned.
As AffineMesh said, the nominal voltage is about 3.6V for lithium ion batteries. My Dell laptop requires ~11V as input. That's about 3 in series (3.6 x 3 = 10.8). So in order to increase the capacity, you need to add 3 that are in series and put them in parallel with your existing cells.
So here's a 6 cell (an = is a cell):
And to get a 9 cell, you add another column of cells: