You will still sweat after laser hair removal

You may notice less odor, however.  The apocrine glands in the armpit and pubic areas are closely associated with the hair follicles, and are probably partially disabled by the laser treatment.  Same with the sebaceous (oil) glands.

So patients with skin breakouts in hair-bearing areas generally experience healthier skin. Hair is designed to disperse the odor from sweat glands, it's a pheromone -evolutionary thing.  So less or no hair may have a beneficial effect in that regard. 

Botox or surgery targets the actual sweat glands, and can limit the amount of sweating.