Ultimately, planetary ring systems are flat because of the oblate (equatorially bulging) shapes of planets, which creates an asymmetric gravity field around the planets. Stellar debris disks don't have these asymmetric gravity fields. They are flat, ultimately, because of the large angular momentum of the disk itself.

Why are the planets orbits flat?

It's thought to have arisen from an amorphous cloud of gas and dust in space. The original cloud was spinning, and this spin caused it to flatten out into a disk shape. The sun and planets are believed to have formed out of this disk, which is why, today, the planets still orbit in a single plane around our sun.

Are Saturns rings flat?

Sculpted by the gravity of the planet and the orbital dances of dozens of moons, the rings are divided into a few major rings and thousands of narrower ringlets. They're also huge: The main ring system spans 300,000 kilometers (180,000 miles), and some fainter rings are cast even wider! But they're also amazingly flat.