As one who has done acupuncture for a few years now, I would say the difference is in how it’s practiced. Reiki or other energy healing treatments are consciously trying to harness and manipulate the “energy” of the universe. That’s what your in there for them to do. And as far as I know, those treatments include incantations and emptying your mind and becoming one with the universe and things of that sort. (Though I could be wrong, I’m just going off memory here from what I’ve read about it.) The Church has a problem with that because it borders on the occultic, trying to manipulate the spiritual realm.
That’s not the case with acupuncture. There’s no spirituality attached to it. (unlike Reiki) We’re not told to empty our minds and harness the energy of the universe or anything of that nature, there’s no incantations. Literally, it takes five minutes, the acupuncturist pokes you in certain spots, depending on your medical issue, and then she leaves.
The one I go to is a westerner, and she was able to “translate” what acupunture does without using New Age terms. It’s basically pressure points on the body that stimulate certain reactions, I don’t know how the science works, but it has helped. I have a genetic disease, and it weakens my muscles, and since going I have had massive improvements. (Though I do know someone who got poked in the wrong spot and it sent her into shock!

Which to me showed those pressure points really do have an effect, and not always one we want.)
So, the reason one is forbidden and one is not has nothing to do with their origins or lack of hard studies proving them, or even the terms they use to describe what’s going on, (like chi and meridians and things) and more to do with their techniques. Harnessing energy and attempting to be one with the universe are forbidden by the Church, and that’s what Reiki and other energy healing practices try to do. Using pressure points to stimulate certain reactions, while maybe not proved scientifically yet, doesn’t fall into that category, so it’s considered ok.
It’s about the spirituality, not the words used, and like I said, acupuncture does not involve any sort of New age spirituality or attempts to empty your mind and harness anything. It’s just an older form of medicine.
That’s my take on it anyway. I will say though, that if the Church investigated and did say acupuncture was spiritually dangerous, I would quit going. (To acupuncture, that is

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