Soup vs. potage

Soups

Basically, it's water and vegetables cut into small pieces, seasoned and cooked until the vegetables are well done, to obtain a soup.
Of course, the basic recipe can be enriched by replacing the water with broth or milk, for example, or by adding various small ingredients: leftover cooked meat, legumes, croutons, etc. etc. The only limit is your imagination, and what you have in your cupboards or fridge.

Potages

Basically, it always starts out as a soup: water + vegetables that you cook, but then there's the difference: you blend it or put it through a food mill, and you get a soup.
A soup that can also be enriched and improved by adding all kinds of ingredients: Cream, grated cheese, a little cornflour to thicken, etc.
The term "potage" comes from the 17th century, in the sense of "vegetables cooked in a pot", and then gave rise to "potager", a place where vegetables are grown for soup.

As you can see, it's more a question of texture: soup has chunks, while potage is smoother.

Having said that, this boundary is rather blurred, as the 2 words are used for a whole host of very different dishes on restaurant menus, for example, and thus fall outside the usual domain: starter or single dish, hot and savory.
Desserts include fruit soups, for example, which are often a fruit juice or a custard, or both, in which pieces of fruit are placed.
In short, with the imagination of chefs, a soup is just about any dish, made up of a liquid part and a solid part in small pieces, strawberry soup for example.

Other words

In addition to soup and potage, there are other related words, a little outdated nowadays, to designate more specific kinds of soup:

Consommé: In principle, this is a soup in which the water is replaced by a broth, usually beef, which is fairly full-bodied and sometimes with wine added. In ancient times, it was a popular starter at festive meals.

Velouté: This is a soup generously enriched with cream, and sometimes bound with a little cornflour, and very carefully blended or even sieved, all to obtain something very smooth and creamy.

Bisque: This is a soup based on cream and shellfish (lobster, shrimp, etc.).

All these appetizing variations should not blind us to the fact that there's almost nothing simpler to make than a soup at home: water, a few vegetables, salt/pepper and you're done!
And let's not forget that soup has been the mainstay of our meals, if not the meal itself, for centuries (just look at the number of expressions with "soup" in them: soupe au lait, soupe à la grimace, cracher dans la soupe, ...).
It's a dish that's probably as old as humanity itself, and can be found in every culture, in every country. A good thing it has in common with bread.

To sum up: soup is mainly vegetables cooked in water, so there are chunks, whereas potage is a soup that has been blended or put through a food mill.