Headphones are distinct from headsets.
They might be related, and in some cases the same, but they do not serve the same primary purpose.
For example, nearly all headphones will not work as headsets. Making it a synonym seems inappropriate in this case.
A headset requires some sort of microphone input. It's intended to allow communication into the system.
Headphones have no such requirement. They are primarily intended to listen to content.
Many headsets may work as headphones, but most headphones will not work as headsets.