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I am currently creating a traveling merchant character as both my player's background and his profession (adventuring as a ranger class). In the 5e handbook, it states that the merchant variant of the artisan background can start off with a cart and mule, which painted a picture in my mind of me sitting at the helm of a wooden box on wheels, with a rounded tarp hiding all that was behind me.

As I looked for clarification on how much I can hold and how big it was (to see if I could carry my party and my wears) I came across a topic on this site that shed doubt on my vision. "What is the cargo capacity of a Cart versus a Wagon?" brought a realization that there are both carts and wagons in the manual and I am unsure as to which is which.

Upon further search, I came across hints like limitations on how many animals can pull it, and how they become immovable when they hit their weight capacity, but nothing that I could use to differentiate the two. That said, I have known of many kinds of carts and wagons, often having little or no defining features between them. And with no images or dimensions listed in the book, I am left wondering if I should upgrade or not.

So before I continue to spin my wheels, I would like to request the wisdom of this community to help me differentiate what makes a cart and what makes a wagon.

(size, features, uses, etc)

Carts have a single axle (two wheels), wagons have a double axle (four wheels).

The rules provide no detail about the differences between a cart and a wagon beyond what you have likely already found:

$begin{array}{|l|c|l|}
hline
textbf{Item} & textbf{Cost} & textbf{Weight}\
hline
text{Cart} & text{15 gp} & text{200 lb.}\
text{Wagon} & text{35 gp} & text{400 lb.}\
hline
end{array}
$

From this information we can sensibly deduce that a wagon, being heavier, is likely larger than a cart and thus capable of carrying more - but exactly how much more we don't know.

With neither 'wagon' or 'cart' defined more explicitly in game terms we can instead draw from the plain english meanings of these words for further insight. While in common usage these words are often used interchangably,
Wikipedia states:

A cart is a vehicle designed for transport, using two wheels and normally pulled by one or a pair of draught animals. It is different from a dray or wagon, which is a heavy transport vehicle with four wheels and typically two or more horses, or a carriage, which is used exclusively for transporting humans.

This information loosely tallies with the stats we've got for carts and wagons in 5e and so it seems reasonable to conclude that wagons and carts in 5e are similar to wagons and carts in the real world.

There is nothing to prohibit either size vehicle from having a rounded tarp roof.

My understanding has always been that a cart has a single axle while a wagon has 2 or more. Wikipedia and Webster agree with my definition of a cart as being 2-wheeled.

Both of the pictures in your post show wagons. This is a cart: