When a tree comes down, the stump and roots stay behind. Left alone, a stump can take years to rot, attract termites and fungus, and in some species keep sending up new sprouts around the yard.
Stump grinding uses a machine with a spinning cutting wheel to chew the stump and its main surface roots into chips below ground level. It's fast, it doesn't need a big excavator in your yard, and it leaves the surrounding lawn mostly undisturbed.
Full stump removal means digging out the stump and its root ball. It leaves a large hole and takes heavier equipment, so it usually only makes sense when you're building right on that spot, like a foundation, pool, or slab.
For lawns, garden beds, and most landscaping plans, grinding is the practical choice. Once the stump is ground, you can leave the chips to settle, or clear them and fill the hole with soil and lay sod on top.
If you're removing a tree anyway, it's usually most cost-effective to grind the stump on the same visit. Ask for both when you request your estimate.