Torture is to cause pain, to someone and is more often associated with physical torture, though it can also be associated with mental pain. They tortured him for hours, with hot pokers. The pain was pure torture.
Torment is purely emotional, we torment people with words, thoughts, suggestions, etc etc. Or things can torment us emotionally, They tormented her every day in the school playground, calling her fat and ugly. His words tormented me..