Hurt and ache can have very similar meaning and can sometimes be used interchangeably. I will try and give you some examples.
Both hurt and ache convey the idea of sensing pain (dolor). If you cut your finger, you could say either - My finger hurts or My finger aches. The meaning here is almost identical. The verb in both cases is used intransitively, and that is the key point. Ache when used as a verb is used ONLY as an intransitive verb, meaning it has no object. As far as I know it not used as a transitive verb, but hurt is used in that way. You might hurt someone by pinching them, but you would not ache someone. That is the main difference. You said you were interested in the verb, but you might want to know something about the term ache when it is not used as a verb. In the US we often say both my stomach hurts or my head hurts, it is less common to say my head aches or my stomach aches. I think it is most common for an American English speaker to say I have a headache or I have a stomachache in preference to the other examples. This would also be true of a tooth, but I can't think of it being used with other parts of the body, such as hand, foot, leg etc. You would not say I have a footache or an armache. You could use the term heartache, but it would not refer to actual physical pain but rather an emotional or mental distress.
So to sum it up, I think using hurt as a verb is more common... you can't go wrong with it, so when in doubt, opt for hurt as a verb. Don't use ache as a transitive verb. My dictionary classifies it solely as an intransitive verb. The expressions to have a headache, a stomachache or a toothache are used, in my opinion, much more frequently when describing pain in those specific body parts. For other pain I would say hurt although you could say ache and be perfectly correct. I think ache is used most commonly when referring to emotional pain - such as my heart aches for those hungry children. I suppose one could say my heart hurts for those hungry children, but I think its use is very awkward sounding and less desirable.