I honestly don't know why people react that strongly against witches in this manga but I think the difference between witches and magicians is that witches are an illegal occupation. You gotta register and study magic in some school or under some great magician as their apprentice in order to be recognized as someone who handles magic.
And I'm only guessing here but there might be some sort of superstition against witches or some long and dark history against witches. Also I think that magicians are like a corrupt government that monopolize medicine and magic in order to reap the benefits of it
Between a witch and a magician is there actually a difference? They seem to both be able to use magic and make medicine so is it literally just being able to learn officially at a school or on your own?