I can't find any good answers. The Waffen SS was more of a private army while the Wehrmacht was a national army I think. Except that the Waffen SS was given orders by the OKW just like the Heer and they didn't seem to be used in a different way than the Heer. I searched on the internet. What came up wasn't what I was searching for. I also have a book talking exclusively about the Waffen SS, but never talking about the differences with the Wehrmacht. Yes they are like "private" but they seem to have had the exact same role in the war. They were sent to battle like every other unit.
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The hierarchy between the Waffen-SS and the Wehrmacht was quite similar -- as you pointed out, both were commanded by OKW. The main organizational difference was the recruitment source. The Wehrmacht drew recruits from German citizens, just as any other country's military would draw recruits from its citizens. The Waffen-SS drew recruits from the Allgemeine-SS (which had racial requirements on recruits) and ethnic Germans (though the Nazis weren't really able to maintain their racial purity ideals as the war progressed):
(2) Recruiting of foreigners. With the invasion of Russia in June 1941, German propagandists set themselves to the task of changing the whole aspect of the war from a national German affair to a "European war of liberation from Communism". In this way the Nazis were able to obtain a considerable number of volunteers from occupied and even neutral countries, who were organized in combat units of their own in German uniforms and under German training. The original policy was to incorporate racially related "Germanic" people, such as the Dutch and Scandinavians, into the Waffen-SS and non-Germanic people such as the Croats into the Army. When the failures in Russia and other increasing difficulties began to affect the morale of the foreigners, their "voluntary recruitment" became more and more a matter of compulsion and their service in separate national units had to be brought under more rigid supervision. The organization of such units, therefore, was turned over in increasing measure to the Waffen-SS, even in the case of racially non-Germanic elements.
Handbook on German Military Forces, U.S. War Department Technical Manual TM-E-431, 15 March 1945, Chapter 1: The German Military System. (source)
Some Waffen-SS units with notable recruitment sources include:
- 1st SS Panzer Division Leibstandarte SS Adolf Hitler: Originally Hitler's personal bodyguard, its members swore personal allegiance to Hitler.
- 3rd SS Panzer Division Totenkopf: Most of its original recruits came from the SS-Totenkopfverbände, which administered the concentration camps.
- 12th SS Panzer Division Hitlerjugend: Most of its recruits came from the Hitler Youth.
Because of its closer link to the Nazi Party the Waffen-SS was entrusted to act against internal and political enemies whereas the Wehrmacht was to focus on foreign enemies:
- Since the Wehrmacht is dedicated exclusively and solely for deployment against the external enemies of the Reich, the use of the Waffen-SS at home is in the full interests of both the country and the Armed Forces.
Michael Reynolds, Men of Steel: The Ardennes and Eastern Front 1944-45, p. 2 (source)
The Waffen-SS also had different uniforms and tended to receive priority on new/replacement troops and equipment because it was considered more of an elite force. Regarding the priority for troops, for example:
The German High Command has been particularly successful in placing the various types of men where they best fit, and in selecting those to serve as cannon fodder, who are told to hold out to the last man, while every effort is made to preserve the elite units, which now are almost entirely part of the Waffen-SS. The German soldier in these units is in a preferred category and is the backbone of the German Armed Forces.
Handbook on German Military Forces, U.S. War Department Technical Manual TM-E-431, 15 March 1945, Chapter 1: The German Military System.
Otherwise the Waffen-SS and Wehrmacht generally performed the same role during the war. Had Germany been victorious, the differences would have been more apparent after the war as the Waffen-SS would have continued to act against internal and political enemies whereas the Wehrmacht would have reverted to a normal peace-time role for a military force.
If your question is only about their role, ignoring composition, selection, weapons, etc. I'd say that the sole role I can recall that was different compared to the Heer is their role as guards in concentration camps. Check here for further information.