I’ve used all three types of drives: Belts, Chains, and Gears.
This past year was my first experience with a custom belt-driven drive system. The main benefit to belts (and the reason we chose them) is weight savings compared to chains or gears. That said, the tradeoff is that your drive system will tend to be wider (leaving less room inside the robot for other things) as belts tend to take more space than either of the other two options (assuming you’re using 15mm belts, of course. It’s not as bad with 9mm, but you’ll tend to break them more). The other downside to belts is they can be a pain to do maintenance on, since the only way to replace them is generally to remove the sprockets from BOTH drive and driven shafts. You also have to keep stock of the EXACT length belts you need, since you can’t exactly make a custom length belt at an event (or rather, you probably shouldn’t try to).
Chains have the advantage of being easier to work with, and are generally more forgiving to slight spacing errors. This makes them great for teams trying to make a custom drive system or other mechanism, but that don’t have the expertise to get their hole spacing exact. Chains can also be removed and attached without necessarily having to remove any of the sprockets (if you have a master link or an appropriate chain tool), and generally take less space than a comparable belt system. The downsides to chain is it does stretch out over time, requiring either tensioning or replacement; it also (in my experience) tends to break more frequently than belts. As far as weight goes, it generally falls in the middle between belts and gears. I’m partial to using chain (#25 chain, specifically) for mechanisms, since it allows for easy and rapid design modification, but if your team is capable of precision machining, they really aren’t necessary for most drive systems anymore
Gear-drives have a few advantages over the other two; for starters, they’re significantly easier to do maintenance on (and require far less of it). The lack of any tension in a gear-drive means you can simply pull the dead-axle shafts on your wheels and they drop right out without affecting the rest of the system, it also gives them a slight efficiency advantage. Gears also allow you to build the narrowest drive systems by far, as the gears only require ~0.5" of space regardless of how many wheels you’re using (assuming you’re using COTS FRC gears and not some custom solution). The obvious downside to gear-drives is their weight, due to all of the wheel and idler gears (and their associated shafts and bearings); this can be mitigated to a large extent (as gears can often be pocketed dramatically without weakening them too much), but they’ll still tend to be heavier than the other two solutions. Also, because your wheel spacing is determined by the size of your gears (and the size of your wheel gears is limited by the size of the wheel), you either need a LOT of wheels, a lot of idlers, or a combination of large wheels and large idlers. In general I like these kinds of drive systems for defense-heavy games (that don’t require a lot of heavy mechanisms) , or for games with unusual floor geometry where you don’t want to use “Tank Treads”.