The short answer is pretty much nothing. Today they are widely referred to as synonyms. Even by our trusted friend google. Some people consider couches to be more casual and sofas to be more sophisticated. This could be because of their history.
The term couch was derived from the French word “coucher” meaning “to lie down” or “to sleep”. They were small, cozy, and usually armless (see below). This classic couch is similar to what we call a chaise today. They were traditionally used for taking naps, relaxing, and getting comfortable in private.
The term sofa originated from the Arabic word, “suffahs”. These were traditionally wooden benches and sometimes dressed with blankets and cushions. Their primary use was to act as an elegant seating area for guests.
As these pictures convey, we have come a long way from the origination of these staples. These days our living rooms are used both socially and privately which may be why sofas and couches have become one in the same. Some designers do abide by these differences in naming more casual designs couches and more classic models sofas - but overall society does agree with asserting the same meaning to both.
