Two names for one pot
A Dutch oven is a heavy pot with a tight lid that goes from the stove into the oven, for braises, stews, soup, chili and bread. Le Creuset says its French oven is the same pot, also called a cocotte, named after the 18th-century Dutch design it improves on. In the US, Le Creuset sells it as a Dutch oven.
What the name usually tells you: enamel
When someone says French oven, they nearly always mean enameled cast iron: iron coated in glass enamel inside and out. Le Creuset says the enamel resists chipping, cracking and staining, and its light interior makes it easy to watch food brown. A Dutch oven can be enameled too, or bare cast iron like Lodge's classic pot, which is seasoned with oil instead.
That difference changes how you cook and clean. Enamel needs no seasoning, washes with soap and water, and copes with tomato sauce or wine simmered for hours. Bare iron needs drying and a thin coat of oil after washing, and long acidic simmers wear its seasoning.
Which one to buy
Ignore the name and choose by material and size. Pick enamel if you cook a lot of tomato, wine or bean dishes and want soap-and-water care. Pick bare cast iron if you want a pot that takes high heat, can be seasoned again if it rusts, and costs less. For size, about a quart per person is the starting point Le Creuset and Lodge both use.
How these comparisons are made
Questions about this choice
Is a cocotte the same as a Dutch oven?
Yes. Cocotte is the French word, and Le Creuset uses it as another name for its French oven, which it also sells in the US as a Dutch oven.
Can a French oven go on the stove?
Yes. Le Creuset says it is used for searing, sautéing and boiling on the stove as well as braising and baking in the oven.