I looked at the (old) Osprey book on the Waffen SS, and it seems that the uniforms are the same as the Army ones except for the insignia. Is this the case?

The regular wool 4 pocket uniforms are the same yes,except for the eagle,which moves to the left arm and the collar tabs,of course,has the SS system of rank.
The tank uniforms have a slightly different cut to the panzerwrap though,with smaller and more rounded lapels.
Now,cammo uniforms are a whole different story.
J

Outside of the Panzer wrap there were subtle differences in the tunic but nothing discernible in 1/35 scale. And army clothes stocks were often used with the changes in insignia so yes and no.

As the war went on the Kriegsmarine and Politzei among other branches of government also used stocks of army uniforms with only the insignia being different.

After 1942 the Nazi government almost became one morass of field grey with a bewildering array of insignia.

Although if you could read the insignia, it told you exactly where that person belonged. One more aspect of the 'alles in ordnung' drive to pigeonhole everyone precisely into their role in the Greater Reich.

The Waffen SS did have their own version of the service tunic - Slightly different collar. slash type lower pockets, different button spacings , so that a collar and tie could be worn and the shoulder straps were faced in black. Supply, once war began, meant that Army M36 tunics with SS insignia were issued.

Not forgetting the early pre-war Earth Grey uniform which was still being worn by vets in the Polish and French campaignsalong with the pre-war regimental collar patches or without collar patches in France.