Those are used in different context entirely. doux : nice to stroke (like a cat, a sweatshirt, etc). OR mild taste. OR sweet mouelleux (or "mou") : soft texture, as in if you press it, it will dent it (like a cake, a couch)

doux = sweet,soft,pleasant,mild, gentle moelleux = sweet, soft, spongy, creamy, mellow, smooth, fluffy They are both similar.

As your picture suggest, you may refer to wine. About legislation : A moelleux wine is a wine that has a certain content of sugar after a full fermentation. This is between "demi-sec" and "liquoreux" according to the European regulation. A "vin doux naturel" is a wine for which the fermentation process has been stoped by an addition of alcohol. About tasting : You can say that a wine is moelleux when you get a feeling of smooth and velvet in your mouth. This sensation is perceptible on the tip of your tongue. You call "doux" a wine that is sweet.

"Moelleux" is more used to talk about things as pastry or something to eat and "doux" is used to describe a texture, something which is soft as silk or other things as animals..