絨毯 is a general term for carpet or rug, often thicker and pile-woven, while 毛氈 specifically refers to felt rugs, which are thinner and made by matting fibers.
The word is composed of 毛 (hair/wool) and 氈 (felt rug). The exact historical derivation is uncertain, but it likely entered Japanese as a term for felt carpets, possibly through Chinese.