Also, I understand how this product benefit deaf people, but I don't understand how it would be any more useful to mute people than non-mute people.
According to Wikipedia, just 'deaf'.
http://en.wikipedia.org/wiki/Deaf-mute
I'll resist the urge to rant about political correctness.
This political correctness stick is brought up on these occasions to ridicule a world where you can't say certain things any more. But this stuff isn't about this invented construct of PCness - does anyone claim to be politically correct? It's just a pejorative term invented by people who want to still be racist and old-fashioned. It's about respecting people whatever their background and not making sweeping and offensive statements about them based on the colour of their skin or where they come from or their disability, sexuality or gender.
Generally speaking, "deaf" = physically deaf, i.e. unable to hear, and "Deaf" = culturally Deaf, i.e. part of a visual-culture / signing linguistic group. Big-D Deafness does not necessarily imply little-d deafness, nor the other way around. Children of Deaf adults, for example, often learn a signed language before an auditory one, despite being physically able to hear. And sadly, many physically deaf children are still deprived of language until quite late in their development.
And, as always, if a Deaf person tells you something that contradicts me, that person is right.
For more info, see http://en.wikipedia.org/wiki/Deaf_culture
Colored can hardly be called derogatory, just ask the National Association for the Advancement of Colored People: the NAACP.
Beyond simple political-correctness, though, the most important thing is to understand how our language affects people. A major problem with "Deaf-mute" (and also with "colored" and "oriental") is that these terms are oversimplifications at best, and drastic misrepresentations at worst. They're not just words - they represent actual cultural misunderstandings.
With the Deaf, for example, the core misunderstanding is usually thinking of deafness as a medical disability. It comes as a surprise to most people that the Deaf do not generally consider themselves "disabled", but instead possessing a different human experience. This is difficult, I think, for a hearing person to imagine if they have not met and interacted with a Deaf person.
Anyways, that was a very long-winded way of saying that it's not just about finding the most politically-correct term to use - it's about finding terms that describe the world as it actually is.