Shades of difference. Most would say that "alternative" medicine can be used "instead of" conventional approaches, and involves whole different methods of diagnosis and treatment. "complementary" methods are more often meant to be used along with conventional treatments, sometimes to ease side effects or to bring in aspects the conventional treatments may not include (such as spirituality, etc).
Probably semantic. They are both terms used to identify applications of treatment outside of the usual mainstream of allopathic medical treatments (drugs, surgery) but those margins are blurring. Traditionally complementary meant adding other approaches like acupuncture, nutritional focus to alloopathic practice and alternative defined those disciplines outside of allopathic medicine. Again margins blurring.