Could be another root deeper down pushing soil up, pushing up your driveway.
I don't know if you live in a colder climate but you could have frost heave. Water is going down in the old crack, freezing and expanding resulting in the driveway to heave up. It's important that when you start to see cracks in your driveway or sidewalk that you seal them right away to prevent this from happening.
If you do decide to cut more roots be very careful not to weaken the tree's support. Some tree's roots grow very close to the surface and when you cut them the tree can become unstable. A few years back in my neighborhood, a guy was replacing his sidewalk and he cut some tree roots back to have a nice level walk. Six months later the tree came down in a light windstorm and landed on a passing car, killing a
4-year-old girl in the car. I lost track of the story but I believe the guy ended up going to jail. It was proven that his actions of cutting what he thought were just a few roots undermined the tree's support. Be careful cutting roots.