Assuming all things equal except for string gauge, higher gauge means higher tension, which in turn means a stiffer string a greater percentage of the plucked energy is put into higher-order harmonics. Tone-wise, this means that strings with higher tension will be brighter, while those with lower tension will have better bass/low-mid presence.
Playability changes as well - lower gauge strings are more flexible and easier to fret, but it's also easier to get fret buzz, leading to higher action or learning to play with a lighter touch. Higher gauge strings are less flexible and harder to fret, but also make it harder to produce fret buzz, leading to a lower action and no change in finger/pick attack. My description really makes the difference sound much greater than it really is (unless you're comparing dental floss and steel rebar).
All that being said, gauge more of a guideline between manufacturers and not quite as descriptive as one would hope. Because of various construction techniques and materials used, you can find a .100 from one manufacturer that has greater tension and/or is more stiff than a .105 from another manufacturer.