You will get more exp on every further difficulty.
As people said, on epic you get -%40 primary resistances along with bleeding and vitality. Rather than that mobs will slightly stronger for your lvl ofc.
About loots rares(blues) will drop more than normal diff and legendaries(purples) will now able to drop.
4 new bosses will also appear from now on epic diff. Talos and Boar Snatcher at Act 1, Manticore at Act 2, Dragon Liche at Act 3. Manticore and Dragon may not drop good things due to their low chance drop rate but Talos and Boar Snatcher can be worth to farming for early stages of epic.
About legendary this is diff where game becomes hard especially last parts of act 3 and almost whole part of act 4. You get another -%60 resistances and mobs become even harder.
Drops will be even better than other 2 difficulties and you will able to drop +45 lvl legendary items which you cannot even get from anywhere at epic.
Hydra boss will now appear which only appears at legendary diff and he has very good drop table. If you want gear(you will definitely want for this hard diff i believe) this is the boss for you. He has chance to drop every legendary item(except gloves) between lvl 36-46. So for gloves you may want to farm them from act 2 telkine.
Once you done your gearing from Hydra or bored from it then start to farm Hades on legendary, who has best loot in the game cause he can drop ''any'' legendary item which includes 50-55 legendary items. Secret Passage is another option but i really wouldnt recommend it cause it drops many easter egg items and can be very time consuming for melees cause of new added cage mobs.