I'm looking for a sander and as I browse through the various offerings I can see that most manufactures offer both a sheet sander and a random orbit sander. Aside from the shape of the sanding surface, are there any functional differences between the two? What scenarios does each type excel at?
In addition to Rob's answer, the motion of both sanders is a bit different.
Sheet sanders tend to move back and forth in a straight line, with small variation side to side.
A random orbit sander has an eccentric cam mechanism that makes the sanding disc move in a circular but not round fashion. I'd best liken it to the old spir-o-graph toy- it's 'orbiting' around the central point, but it's not a simple circle. You get less noticeable scratches this way.
As to the difference in application.. sheet sanders excel at getting into corners and edges, but can cause grooves because the sheet only moves in one direction.
An RO sander is great for wide open spaces, where if you use a successively finer grit, you end up with a nice smooth surface. But it doesn't get into inside corners very well. My RO sander also has holes in the pad, which facilitates sanding dust removal with a shop vac. (I'm not sure if modern sheet sanders have a dust collection port, but none of mine do.)
Sheet sanders usually take 1/4 or 1/2 sheet of sandpaper held in place with 2 clamps, though some can also take hook-and-loop paper. As Graphus pointed out, most models today have dust collection. Many models come with a hole punch in order to perforate standard sheets of sandpaper with the appropriate pattern of dust collection holes, but for some you must buy the hole punch separately. Sheet sanders themselves are relatively inexpensive and the paper is also very inexpensive on models which use normal sandpaper, compared to sanders that use hook-and-loop sanding media.
Random orbit sanders typically use round sanding disks held in place with hook-and-loop (velcro), and have holes in the both the sanding media and the sander's pad for dust collection. A random orbit sander costs the same or more than a sheet sander with more or less the same specs, and the sanding discs cost more than normal sandpaper. You have to be somewhat careful when buying media, because there are 2 "standard" but basically incompatible hole patterns, 5-hole and 8-hole. Although 5-hole sanders are not as commonly sold today, you can still find 5-hole sanding discs. Festool has its own proprietary hole patterns, called Jetstream and Multi-Jetstream. Mirka has its own line of Abranet sanding mesh which, as mesh with many small holes already, doesn't require any particular hole pattern and can attach to any hook-and-loop sander.
Dust collection not only reduces the amount of dust in the air and all over the place, but it also allows the sandpaper to work more efficiently by preventing a layer of wood dust from building up between the sanding media and the workpiece. The resulting airflow across the sanding media's surface also helps against excessive heat buildup, which will melt and destroy the plastic hooks (for the hook-and-loop fastening) on the bottom of the sander's pad.
As TXTurner and LeeG pointed out, the two types of sanders also produce different scratch patterns. Sheet sanders can reach into corners which a random orbit sander cannot do with its circular pad.
Sheet sanders use a linear or orbital (circular) motion, and some can be switched between the two. Random orbit sanders use an eccentric circular motion which is intended to create a less visible scratch pattern, but some higher-end models can also be switched into a gear-driven orbital mode for more aggressive sanding.
Random orbit sanders rotate and oscillate to give scratch pattern that is harder for the eye to discern. Sheet sanders typically just have some random oscillation. RO sanders can be more aggressive.
Both sanders can support dust collection (see Festool), but typically sheet sanders do not have provisions for dust collection, and RO sanders almost always do.
The larger sheet sander (1/2 sheet) are great for finishing a large, flat surface. Sheet sanders also work well for sanding into corners, something you cannot do with a round RO sander.