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

Both other comments are quite literal about traveling, but here are two metaphors, too:

On the way is when you're actively progressing. You've got a goal in mind and are traveling towards it.

On the road doesn't really mean progress, but discovery. As an analogy "on the road" implies meandering, getting around, and experiencing things. It's a relatively long duration with no particular goal.

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.