[Lil' Lavery]

Assuming that there is one motor per wheel, and that all motors used are identical:
With a four wheeled system you get between 50% (when translating along the axis of a wheel) to 71% (when translating 45º from an axis of a wheel) of your total force of your four motors. Therefor, you are harnessing between 2 to 2√2 (~2.83) motor's worth of force.
With a three wheeled system you get between 47% (when translating along the axis of a wheel) to 68% (when translating 30º from an axis of a wheel) of your total force of your three motors. Therefor, you are harnessing between √2 (~1.41) to 2 motor's worth of force.

The three wheeled system has a lower range of efficiencies, but only very slightly, but has less total force (due to the fact it has one less motor). This reduction in force results in slower acceleration and less pushing power.

Four wheeled frames additionally are able to be spaced further apart, resulting in a much larger support base (as you can cram the wheels into the edges of the size requirements). This results in a few things. The biggest advantage is the increased stability gained by this base. Additionally it creates a bigger "lever arm" for the wheels, as they are further away from the center of gravity of the robot, resulting in them being able to apply more torque to the bot*, which increasing your ability to rotate (or resist being rotated).

Four wheeled frames are also easier to construct from "standard" kitbot frames (or similar), as they only require 90º angels, as opposed to 120º and/or 60º angles needed for a three-wheeled platform.

Four wheeled drive bases often weigh more, as the frames tend to be larger, and you need an additional wheel and motor.


*Please note, this is torque to the bot, not the wheel's torque. It won't increase your pushing power, but rather the bot's ability to spin.