Both are great choices. Massage therapists deal with muscular issues that affect you and there is nothing better than hands-on healing. Some times massage has a Western mentality to it, such as if your back is hurting it is because of the muscles in your back, while they also learn a lot of other things such as stretching techniques and use of essential oils that will all contribute to your pain relief. Acupuncturists are very precise and spend a great deal of time learning about very specific points in your body that affect you. They are very Eastern in mentality, such as, you are waking up at 3AM because your meridian that is most active at that hour is not balanced. Acupuncture might not feel as good as massage but the effects are real and documented and this practice of medicine has been around for thousands of years and is proven, which is what the acupuncturists would argue. Massage therapists would argue that massage predates Christ and that there is no substitute for a hands-on approach, working your muscles out. I would suggest you try each of them out, just once, to see which one has a greater effect. But my best advice to you would be to go and see a chiropractor -- most of the time he/she will take your insurance and have either an acupuncturist or massage therapist working with him/her to give you the best experience in pain relief, with a doctor's advice always ready and the ability to get x rays or other tests that either a massage therapist or an acupuncturist would be unable to order.
I have seen chronic trigger points causing patients to experience undue pain and massage won't help it but stick a needle in there and it goes away immediately. Massage therapists some times refer to acupuncturists if they know they are unable to treat you, and likewise for the acupuncturists referring patients to get massage, because there are some conditions that only massage will cure and there are some conditions that only acupuncture will cure. It really depends on the source of your pain -- whether it is a chronic pain or caused by a recent accident, or whether it is because of your job, etc. You could try to find a person who is both an acupuncturist AND a massage therapist -- I know many, and I am both myself. Look around and you'll be surprised how many people are out there who are willing and experienced and want to help you with your pain relief.