So as it turns out, one of the little rocks I found a long time ago and thought to be jasper eventually turned out to be a carnelian. What's the difference?

While jasper maintains it's opaque surface no matter how thick or thin it is, carnelian becomes more translucent the thinner it gets. Some carnelians (usually red ones) can be confused for a jasper unless broken into smaller pieces, cut very thin or if a chunk of it is held close to a light source; if a carnelian has thinner edges, they will be translucent.

Which is exactly what I did with this one after giving it a thorough look and noticing how different it looks compared to all other jaspers.