1. Intransitive verb: They mean the same. My back hurts. My back aches. 2. Transitive verb: "ache" is not transitive! You can only say: If you bend too hard, you'll hurt your back. The shoes hurt my feet. (You cannot use "ache" here) "Hurt" also means "injure". 3. Noun: Ache: If something hurts or aches, then you have an ache. You do not have a "hurt"! (nonsense) Hurt: refers to emotional hurt only. The hurt from the break-up is still strong. That's all for now.

In (American) English, people often speak of "heartache" (heart + ache) meaning that they are missing people the love or something similar. "Ache" often means a longer term pain (headache, stomachache, toothache, etc). "Hurt" often means a shorter term pain: (I hurt my finger, I hurt my toe, that hurt my feelings).

An 'ache' could be described as a dull pain, that is not necessarily less painful, but a constant pain, usually from within a limb or other part of the body. On the other hand, "my knee hurts" could describe a sharp pain.