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.
Answer: Double (x2)
What's not said is important.
This may be an oversight by the 5e's dev team, or something 'assumed,' but a cart can (by house-rule/common sense) be drawn by only one animal, whereas a wagon can be drawn by 2.¹
The formula is 5× the carrying capacity of the animal drawing the vehicle. So, for a cart via a donkey, it's:
2100 lbs. − 200 lbs. (for the cart itself) = 1900 lbs., as @Miniman's answer states.
Except a Wagon would allow for two donkeys... meaning:
4200 lbs. − 400 lbs. = 3800 lbs.
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 50 sq. ft. and the oak a 400-lb. 'wagon' couldn't withstand more than ~4000 lbs. of direct pressure (let alone iron axles and reinforced spoke wheels) although at that weight you would need 6 mules in reality.
Adding to what others have said, a cart has two wheels (one axel) while a wagon has four wheels (two axels) so it stands to reason that a cart can carry half the load of a wagon, in terms of weight.
Some quick googling roughly supports this idea, and puts wagons at a potential 2500 lbs. while recommending 1600 lbs. max. So a cart would be 800 lbs. or roughly 4 medium creatures with gear.
As others have mentioned, pulling that weight is entirely up to the creature(s) pulling it.