Its probably a final drive gear mesh issue (the mesh from secondary shaft to the large final drive gear that drives the diff). This could be incorrect tooth manufacture/ profile or more likely gear-gear misalignment. In the second case, this may be a function of a suspect bearing or some other factor causing the gear to run out of alignment (housing, diff casing maching error).
In scottings case it sounds like the side gears and pinions in the diff had worn or become damaged, causing the diff to lock intermittantly, which tends to lock the steering momentarily. My old car did this before the diff gears started to fail to engage with each other as they had gradually worn through their seats on the diff housing.
Regular oil changes help, as does not giving it the beans in the snow/ wet from cold and inciting lots of cross axle speed difference. Replacing the diff internals would fix this but would not affect whine, as this is gear tooth mesh (possibly as a consequence of a bearing) related. I would think its cheaper if you look at labour costs to swap in a reconditioned box than get into replacing bearings (taper bearings on the diff so need re shimming) if thats what it is.
Anyway, back to the original post, I would drop the oil and check for any suspect metallic particles, check for excessive rattle, lumpiness, noise or play in the diff and final drive (rotate the driveshafts by hand, wheels off the ground). If all seems ok then either a) live with it and keep your fingers crossed (one of my old cars had a final drive whine for all 165k miles of its life) or b) incite it to get worse/ break with plenty of abuse (or run without oil for a few miles...). Alternatively start looking for a cheap box now and get it swapped when you do a clutch change.
Definitely have the dealer look at it properly though and consider an external assessment (rac/ aa maybe) as its a suspect part on a major and expensive component. The whine is not normal and is an indication something is not correct, one way or another.