What is the difference between a 3 and 6 cell battery in a laptop? Does it really provide that much more power? Will I see a substantial difference in upgrading to a 6 cell?
tl;dr: more cells = longer running time.
All batteries are composed of one or more chemical cells, each cell typically produces a voltage of between 1.2V and 1.5V depending on the chemistry. The capacity of a cell is usually measured in Amp-hours (Ah) - though in reality this depends on current drawn, temperature, age and other factors.
To produce the voltage needed by the device, you add cells in series. Three 5Ah 1.2V cells in series might produce 3.6V for 5Ah.
To increase the running time of the pack, you add cells in parallel. three lines of three cells (nine total) might produce 3.6V for 15Ah
The main differences will be weight, run time, and possibly size.
The 9 Cell battery can store more charge than the 3 Cell version. A 9 Cell battery will also tend to be heavier, and depending on the laptop may also be physically larger. This is a generalization though, as the individual cells used in each battery may not be exactly the same, so the batteries can only be roughtly compared this way.
For more information you will want to look up the capacity of the battery (in Ah) and compare those two numbers.