What’s the difference between “way” and “road”?

There really isn't a consistent difference between them.

Some municipalities may have a scheme, but it is likely local to that municipality.

As a general term, "way" is just an archaic word for a road. If you want to be somewhat poetic, you might use it. If you are going for straight clarity, you'd use road.

It depends on the context and the dialect of English being spoken.

They live right down the road and They live right down the way are approximately the same, although "way" in that locution is not used as frequently as "road" is, at least in my neck of the American woods. It was used more often by people in my grandparents' generation.

If I saw two street names: ________ Road and ______ Way, I'd assume the Road would take me from one town to the next, or at least across town, whereas the Way might be only a block or two in length.

To be "on the road" means to be traveling over an extended period of indefinite length, but a length which would seem a longish time to both speaker and conversation partner, in a given context.

I've been on the road for an hour in heavy traffic.
I've been on the road all day.
I've been on the road this week, visiting customer sites.

To be "on the way" means to be in transit.

Hello, Sis? I'm on the way and will be there soon. Traffic is heavy, and I'm running late.