Yes, I would largely agree with others so far. But there is a key difference, surely: whine specifically describes making a high-pitched sound, whereas whinge is to complain in a rather pathetic manner. So though a whinge may be uttered in a whining tone, a whining tone is not the same as a whinge: it is used synecdochally. Personally I would pronounce them as winj and wine.

I will be interested to hear more detail about the roots of the words, since an initial check also suggests that they are not quite so identical as originally suggested:

whine (hwîn, wîn)

v., whined, whin·ing, whines.
v.intr.
1. To utter a plaintive, high-pitched, protracted sound, as in pain, fear, supplication, or complaint.
2. To complain or protest in a childish fashion.
3. To produce a sustained noise of relatively high pitch: jet engines whining.
v.tr.
To utter with a whine.
n.
1. The act of whining.
2. A whining sound.
3. A complaint uttered in a plaintive tone.
[Middle English whinen, from Old English hwînan, to make a whizzing sound.]

whinge (hwĭnj, wĭnj)
intr.v. Chiefly British., whinged, whing·ing, whing·es.
To complain or protest, especially in an annoying or persistent manner.
[Dialectal alteration of Middle English whinsen, from Old English hwinsian.]

The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2000 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved.