I came upon this thread late. I will define modern medicine as allopathic medicine (and incorporate osteopathic as well) vs. homeopathy, naturopathy etc.
Allopathic medicine's one big strength over all the others is its rigorous application of deductive thinking in diagnosisng a disease. It's one big weakness... it sucks when it comes to managing chronic illnesses (things like fibromyalgia, chronic fatigue, chronic pelvic pain etc. etc.) and I see the other fields as a complementary adjunctive therapy.
For instance, if a 55 year old woman has a headache on the left temporal side, an allopathic doctor will ensure she doesn't have temporal arteritis (inflammatory disease of the temporal artery that if left untreated can cause blindness and is not uncommon as a cause of temporal headaches in 55 yr old patients) before prescribing headache drugs. Now, if the same woman referred herself first to a homeopath, they would not be as vigilante. Now if the same woman has chronic left sided temporal headache due to chronic migraines, accupuncture, biofeedback, nutritional therapy with magnesium and b complex vitamins may be just as successful as the allopathic's treatment.
In addition, only allopathic medicine (osteopathic as well) will conduct the clinical trials in an attempt to ascertain the efficacy of their treatment rather than basing it on empirical reasoning and case testimonials by patients(As someone mentioned, our body has the amazing capacity to heal itself).
Allopathic medicine as we know it is one of the newest "scientific fields" around, which is why it is still experiencing birth pains.
Naturopathic , homeopathic medicine , herbal medicine etc. has been around and practiced essentially the same for centuries.
The practice of medicine using rational or deductive reasoning (which defines allopathy vs. homeopathy based on emperic evidence) has been around since Hippocrates, of course, but look where "rational" medicine got us until just recently? (Blood letting during shock , lobotimizing gay men and aggressive women etc.) But also look at its astounding succeses...the erradication of small pox.
When I say the modern allophathic medicine is new I mean how it is now based on gathering large amounts of data and clinical trials rather than pure "deductive" reasoning and this has only been around for less than a century.
In other words, I think the human body is too complex for one field of medicine to claim monopoly about how to fix it.