Use of the new S variant weapons DOES count towards challenges!!
I've just done a test with the M-8 Avenger S weapon which i got from a gold card.
I ran through the tutorial mission as a test only.
before tutorial: M-8 Avenger was at 34326/140000
after tutorial: M-8 Avenger was at 35464/140000
thus confirming that the use of these weapons does allow you to progress your challenges for these weapons where before you would have had to use the common weapon to progress.
Stat comparison for a RANK I M-8 Avenger S compared with the common M-8 Avenger at rank X is:
M-8 Avenger X
Damage 60
Rate of Fire 525
Max Clip Size: 30
Max Ammo 656
Accuracy 65
Weight 13
M-8 Avenger S I
Damage 75
Rate of Fire 525
Max Clip Size: 30
Max Ammo 525
Accuracy 52
Weight 25
So there is an advantage to using these weapons if you are going for the challenges i'd say. marginal yes, but if you get a few turns of these cards during the course of your challenges anyway, then you are more likely to use them and be able to complete your challenges.