Increasing the difficulty will increase the amount of damage that monsters do, give them extra health, and reward you with more gold and experience. Torment difficulties I - VI, however, also give you an better chance to find a Legendary item, and that chance scales up with difficulty. Here's the breakdown for most of the difficulty changes that's in our game guide. Remember, monsters scale with your character, so you'll normally be fighting monsters your level, or the party leader's level in a multiplayer game.
Normal: +0% Health, +0% Damage, +0% Gold Find, +0% Experience
Hard: +200% Health, +130% Damage, +75% Gold Find, +75% Experience
Expert: +320% Health, +189% Damage, +100% Gold Find, +100% Experience
Master: +512% Health, +273% Damage, +200% Gold Find, +200% Experience
Torment I: +819% Health, +396% Damage, +300% Gold Find, +300% Experience
Torment II: +1311% Health, +575% Damage, +400% Gold Find, +400% Experience
Torment III: +2097% Health, +833% Damage, +550% Gold Find, +550% Experience
Torment IV: +3355% Health, +1208% Damage, +800% Gold Find, +800% Experience
Torment V: +5369% Health, +1752% Damage, +1150% Gold Find, +1150% Experience
Torment VI: +8590% Health, +2540% Damage, +1600% Gold Find, +1600% Experience