Many things can go wrong inside a differential. Although the hints are often subtle, most impending failures give fair warning in the form of noise.
Several situations can create ring-and-pinion noise. If the gears have been quiet and begin to howl, they are probably worn or wearing. If the gears howl during deceleration only, it's possible that the pinion-bearing preload has loosened. Howling under acceleration at all speeds indicates that something in the differential -- gears, pinion or carrier bearings -- has worn or no longer keeps the gear alignment correct. If the gears howl while accelerating over a certain speed range, but not all speeds, it's likely that the gears are worn due to lubrication failure or overloading. When a newly installed gear set howls, suspect the design or setup.
A common problem is worn carrier bearings, as indicated by a low-pitch rumble above 20 mph. On vehicles with C-clip axles, the noise may vary while negotiating turns. Worn pinion bearings can cause whirring noises at all speeds, under deceleration and/or acceleration. Pinion bearings tend to whir, rather than rumble, because the pinion is turning several times faster (depending on gear ratio) than the carrier. Badly worn bearings can also cause howl if they do not support the gears correctly.
Worn wheel bearings can be difficult to determine. A very bad wheel bearing typically makes itself heard with great clarity; it's the bearing that is going bad, but not destroyed, that is hard to find. Turning back and forth from hard right to hard left can identify the culprit; however, I've been fooled by right-front wheel bearings that make noise when turning right (which heavily loads the inside-left-front wheel bearing, but also loads the outside-right-front bearing).
One common situation that may not make any noise: The pinion spins, but the tires don't rotate. Broken spider gears can render the differential immobile, and usually make a loud, crunching sound as they make their final departure. A broken ring gear will allow the differential to propel the vehicle for about eight feet at a time, then bang or grind as the section with broken teeth tries to engage the pinion. Depending on ratio, a broken pinion tooth (or teeth) will clunk about every two or three feet.
A broken axle is easily determined. After it breaks, a C-clip design axle can be pulled out of the housing without unbolting anything -- or may even find its own way out. On many bolt-in-design axles, the wheel will give the broken axle shaft away by cambering in at an angle.
A high spot on a gear tooth may sound similar to a broken gear, but will only make noise while accelerating or decelerating, since the spot appears on just one side of the offending tooth. A high spot on the ring gear will make a heavy clicking sound about every eight feet; a high spot on the pinion makes noise every two or three feet and is much more pronounced due to its higher frequency.
Whether large or small, differential noise is telling you something. Listen carefully! If in doubt, pull off the cover or remove the third member for a closer look. Catching a bad part before is ruins others is definitely worth the effort.