Those two verbs are very close in meaning, so it's a bit difficult to explain. It's often used in different contexts, but hopefully this may help!
To hurt (someone or something) can refer to something physical or emotional. Someone or something is directly hurting someone or something. For instance, after a break up, someone may say "he really hurt me." Or something inflicting pain on someone may cause them to say "Ouch! That really hurt!" It can also refer mostly to areas of body parts like when someone says, "my head hurts."
To harm (someone or something) can refer to something displeasant to someone or something's wellbeing or causing a negative effect on them. Sometime's it's not meant to be as direct nor intentional as hurt. For instance, when we talk about the environment it's often said that, "Greenhouse gases harm the environment."
These are just a couple of examples out there. See if this page helps explain it a bit more? http://www.bbc.co.uk/worldservice/learningenglish/grammar/learnit/learnitv191.shtml