Elevation is the biggest difference. New Army is 12,300', Cottonwood pass is 11,100', but has some longer stretches without any water sources.
If you take the route from chicken Spring lake that just skirts lower soldier lake and then follows rock creek trail, it's about 5 miles from chicken spring lake to the first spring just before the new army pass trail. This route overall to the rock creek crossing is half a mile longer than taking the PCT but it's very scenic, and 5 miles of undulating/downhill on easy trail took me about 90 minutes. From a time and difficulty perspective, is it materially different from the waterless stretch over new army pass?