The debate over heat versus cold is an old one. Cold is still the number one choice for an acute injury--that means during the first two or three days. This helps slow down the amount of swelling and inflammation at the site of the injury.
Heat after that can help carry away waste products and debris that's building up from the inflammatory process. Heat can also relax muscles, thus reducing pain from spasm. For chronic injuries, aches and pains, patients are often advised to use whatever makes them feel better.
For some that's a heating pad. Others still prefer a cold pack. Neither one should be left on for more than 20 minutes at a time. So to answer your question: who's right? You both are: ice for acute injury, ice or heat for a chronic problem like back pain.
And to throw a monkey wrench in at the end: movement, physical activity, and exercise are still the best treatment for chronic low back pain. So for the best results, once you've used your heating pad, get up and go for a walk or engage in some other form of exercise.