This word is believed to have been used in a number of different ways historically, with meanings associated with "silence" for 800 years or longer.
By the 14th century, the term was being used to describe a "person who does not speak," according to Etymonline.
The associations of identifying individuals as silent makes this term offensive due to both inaccuracy and the implications of being without a voice.
The National Association of the Deaf provides this clarification in the context of the term "deaf-mute":
This label is technically inaccurate, since deaf and hard of hearing people generally have functioning vocal chords. The challenge lies with the fact that to successfully modulate your voice, you generally need to be able to hear your own voice. Again, because deaf and hard of hearing people use various methods of communication other than or in addition to using their voices, they are not truly mute. True communication occurs when one’s message is understood by others, and they can respond in kind.