In the D&D 5e Player's Handbook, on pages 155-157, is the Mounts and Vehicles section. The only statistics that they give are Cost and Weight. Does anyone know how much a Cart can carry as compared to a Wagon?
The amount that a vehicle can carry is actually determined by the animal(s) pulling it. From page 51 of the Player's Basic Rules:
An animal pulling a carriage, cart, chariot, sled, or wagon can move weight up to five times its base carrying capacity, including the weight of the vehicle. If multiple animals pull the same vehicle, they can add their carrying capacity together.
So a donkey, the cheapest animal, has a carrying capacity of 420 lb. When pulling a vehicle, it can therefore carry 2100 lb. If that vehicle is a cart, weighing 200 lb, that leaves 1900 lb of cargo capacity. On the other hand, if that vehicle is a wagon, weighing 400 lb, that only leaves 1700 lb of cargo capacity.
What's not said is important.
This may be an oversight by the 5e's dev team, or something 'assumed,' but a cart can (house-rule/common sense) be drawn by only one animal whereas a wagon can be drawn by 2¹.
The formula is 5x carrying capacity of the animal drawing the vehicle. So, for a cart via a donkey, it's 2100lbs - 200lbs for the cart itself totaling 1900lbs like @Miniman's answer. Except a Wagon would allow for two donkeys... meaning 4200lbs - 400lbs, or 3800lbs.
Now mind you this isn't RAW, but any good DM worth his salt might see this being the reason why wagons and carts are listed separately.
¹ - Some might argue there could be 'better' or more expensive wagons that could draft more than one team of horses. i.e. 4, 6, or even 8. But having all that extra pull weight is hard to utilize when your surface area is maybe 50sqft and the oak a 400lbs 'wagon' couldn't withstand more than ~4000lbs of direct pressure (let alone iron axles and reinforced spoke wheels) although at that weight you would need 6 mules in reality.