I have a very early HD650 (black driver, original pads) which sounds 'veiled' to some and 'dark' to others.
All stock... so including foam in front of the driver and the foam in the back.
Yes... used them without foam (and stocking as protector) and with the the little foamies on the rear removed.
While clarity improved the SQ didn't (for me) and reverted back to stock and used other means of 'improving' my HD650.
Also bought some spanking new pads.
These are 'firmer' and seem to be made from different type of cloth as well (feels and looks different, find the older pads more comfy so older ones stayed on.
Also had a spanking new (white/silver) HD600 with mint pads.
My findings (all measured on the exact same rig on the same day)
Old HD650 with
old (original) pads and the
exact same driver with
new pads.

It clearly shows (at least to me) the difference in FR.
Funnily enough while the original pads are 'darker' and perhaps 'veiled' to me with the newer pads they sound (and measure) a lot 'fresher' and without a 'veil'.
Same driver ... just different pads.
So is the pad material the culprit, after all they do feel and look different..
Old pads and
new pads but compressed (with extra force) to the exact same thickness as the old pads are compressed (by the clamp force of the HD650 itself)

Yep ... sound the same and measure the same. New pads clamping force is untolerable, old pads ... very comfy.
So how does the old
HD650 with old pads compare to a
HD600 with original (new) pads.

Yep, sounds about correct, the HD600 is considerable brighter and more 'flat' compared to the old HD650 (old pads)
Now the
old HD650 with new pads vs
new HD600
Yes ... indeed the differences between HD650 with new pads (= close to new HD650 ?) and HD600 are pretty small.
Like Hans described just a very small difference in the lows and upper treble.
So to me the whole 'veil' thing can be attributed to FR related things.
THD and CSD is similar for HD650 old and new pads ...