It's not a synonym but quite hard to explain. Doux means really pleasant on touch. Moelleux it's not easy for me to explain with my poor english. You can translate like something which is not hard (soft). Like the cake. I hope that help you.

"Doux" is employed for fabrics, like soft to the touch. Moelleux ( as in not rigid ) is more for material things like pillows, beds, sofas stuff like that. And for food too ( spongy food ).

It might help to remember that "la moelle" (or more specifically, "la moelle osseuse") means bone marrow. So to be "moelleux" is to have that kind of spongy, compressible quality.