The Pragmatic Fallacy
Informally known as the 'It worked for me' argument, the Pragmatic Fallacy is one that is very commonly used by Alt-med proponents and is based on post-hoc reasoning. "I used homeopathy and my ailment went away, therefore everyone should use homeopathy too". Science doesn't come to conclusions in this way. Your assumptions that an action caused an effect has to be demonstrated repeatedly and under controlled conditions to determine if it is actually the real cause, and not just a coincidence, or, if it only played a minor and perhaps unnecessary part in the causal chain. Of course, even if it could be demonstrated that it worked for that one individual, it does not follow that it will therefore work for others.
Equally in what sense does it mean to say something "works"? It could mean anything from 'the customer is satisfied' or 'the patient improves' to a total cure. Most diseases have cycles of improvement and alongside that simply making someone happier or more relaxed plays a big part in their perceptions of disease - especially pain. Alt-med practitioners often commit time and attention to their patients that hard-pressed public health doctors may be unable to do. We all know someone who swears by their mother's cold remedies because they worked for her, so you should try them too! But if the "alternative" treatment is not the cause of any improvement, then others who seek the same treatment will commit resources, time and often money in pursuing something that is not going to do them any good or may not work as well as other, perhaps less intrusive or resource-hungry, treatments. They may also forgo treatments that do work for something that at best is harmless but at worst downright dangerous - see Black Salve.
Of course, individual anecdotes are not a substitute for good scientific studies. The old adage is "the plural of anecdotes is not data". Anecdotal evidence may be a good starting point for further studies but that is all it is. The claims still need to go through testing, analysis and good quality study to be considered as worth committing limited resources to, or to offer patients in place of existing better treatments.