They're both discomforts. But a pain might be (though not always) worse. Pain is something acute or chronic, that you feel as a result of something. Like surgery. There is pain from the incision and for a few weeks after. Usually after a time, that pain goes away. Often, only medication helps pain. There are different meds that can hep pain. But, trying things like acupuncture, hypnotism, chiropractic and many alternative therapies can work as well. After all, these therapies were around and extremely successful for years before modern medicine.
An ache can also be a chronic or acute discomfort. Usually though, an ache is the result of something more underlying, like body aches from the flu, muscle aches from a bad back. These aches can, and often are, JUST AS BAD. I never realized that aches and pains could be so bad until I developed lupus and fibromyalgia. Lupus gives you joint PAINS and Fibromyalgia gives terrible aches. Think of a toothACHE!
I guess when you really come down to it, since both are discomforts, sometime mild, but sometimes severe; BOTH pain and aches are simiar in how they affect you. The difference is in how you EXPERIENCE them.