Proper use and purpose of mouse weights

This feature is mostly aimed at gamers and gives better control over the mouse.
It's also a matter of personal preference.

With a heavier mouse one has to exert more force to move it, so that slight movements that aren't meant to be relayed to the mouse through the hand get negated. If one has a very slick mouse-pad, which is a feature gamers like, less weight would make it gather less momentum, and vice verso.

The idea behind using extra weight is to avoid overshooting the target. With a lighter mouse the initial reaction may be faster but with less friction desired one may need to backtrack a little to correct the aim, thereby negating the speed gained from the lighter weight.

Some games may require exact motions, so a heavier mouse helps. Some games require quick motions, so a lighter mouse helps.

Depending on the physical attributes of the user, some people have larger and stronger hands. Adjusting the mouse's weight is one way of making it comfortable to use.

Psychologically, some people also feel that a heavier mouse is better class, and a lighter one feels cheap.

Some gaming mouses can be loaded with multiple weights, so are sold with multiple and different weights. By combining the right weights, one can arrive at the exact weight he feels comfortable with in his work or his game.

For example, from Logitech G5 Laser Gaming Mouse Review :

The Logitech G5 has an advanced weight tuning system to fine tune the inertia of the mouse, or even throw it off balance.

The system is comprised of two parts, a transparent orange cartridge, and a tin case for 16 circular weights - half of them are 4.5g, the other half 1.7g. Up to eight weights snap into the cartridge's staggered array to give the G5 up to an additional 36 grams of weight. Gamers can now imitate the weight of AA batteries in a corded mouse, go as light as possible without even using the cartridge, or go somewhere in between.

The G5 36 gram adjustable weight cartridge gives you hundreds of variations on balance and heft - including one that's perfect for your unique gaming style.

Part of the fun of owning this mouse has been experimenting with different total weights and weight "balances" - I finally settled on a featherweight setup with 4.5g weights at the top corners so it "digs" just a little when I push on the mouse, and I might change it again. There's no wrong way to load your mouse; like I said, it's fun to see how just a little more or less weight changes the experience.