One big difference is how they're grown. Microgreen seeds are planted and grown in soil, just like their grown-up garden counterparts. For sprouts, the seeds are germinated in water or wet bags for a couple of days, usually in warm dark places, until they sprout. At that point, they're ready to be packaged and shipped to stores.
The problem is that the growing conditions for sprouts increase the risk of bacterial contamination that causes food-borne illnesses. Since microgreens aren't grown the same way as sprouts, they don't have the same risk. Of course, they still need to be handled properly with food safety in mind, just like any raw veggie or green.
Another difference between the two is that when they're packaged, sprouts include the seed, roots, stems, and tiny undeveloped leaves. Microgreens aren't ready to harvest until they grow their first set of true leaves, and serving them with their roots is optional. It's usually easier to snip them off at the stem.