This answer only deals with American English. We have another answer giving the British definitions.
In their first definitions of each word, Merriam-Webster defines a scaffold as the platform that workers stand on (e.g., when working on tall buildings), and scaffolding as the structure that supports it.
They have a second definition saying that scaffold can also be used for the supporting structure, but they don't say that scaffolding can be used for the platform.
I've certainly heard both scaffolding and scaffold for the supporting structure. And scaffolding for the actual platform feels wrong to me, while scaffold doesn't.