They are close in meaning, but subtly different. Sometimes they could be used to describe the same thing. But wilt has more of an immediate sense to it, like something that is happening now or just happened recently, whereas wither has more of a sense of something that happens over a greater period of time. You can use wilt for people, too, in the common expression that someone "wilts under pressure", that is they become weak or tired or less confident under the pressure of something like a job interview or a test or giving a speech in front of many people.