is there a difference between the terms non verbal, non speaking and mute? I've seen them all used for seemingly the same conditions/to describe the same thing but by different people.
Non-verbal vs non-speaking is mostly a matter of preference, as well as some intra-community differences. You can see this post talk about it, or this one. You will sometimes see it used interchangeably.
Generally the most common type of difference these are used for is that someone non-verbal experiences language problems, and might not be able to communicate with language (including AAC) either fluently or in general. It often implies intellectual disability, autism, or brain damage with cognitive symptoms (rather than purely motor ones).
On the other hand, someone non-speaking can have a mechanical problem, like inability to move or coordinate their mouth, or have had their voice box removed, or some other, usually physical, condition. It often implies something like paralysis (ALS for example), dysarthria, having a tracheostomy, or surviving a head-and-neck cancer.
Some people might use non-speaking over non-verbal to emphasize that they have linguistic abilities, just not the ability to physically produce speech.
Non-speaking might sometimes be suggested as the "better" option because "not everyone can speak, but everyone has language!", which is not always true for those who are non-verbal. Some people don't have any meaningful linguistic capability, and communicate exclusively without language. There are valid and important communication methods that don't rely on it.
These are specifications that aren't used much, but are something I noticed regardless. Most of the time, they're either exact synonyms or pure matter of preference. Non-speaking is also a newer term to my understanding, so someone older might be more likely to use non-verbal.
Many people who have language disabilities, because of their disability, will not differentiate these words. The difference, when it's there, is rather minor in the grand scheme of things.
As a bonus, there is also (rarely) oral vs non-oral. Someone could be oral, i.e. produce sound, but not speaking or verbal, because the sounds do not form speech or maybe aren't communicative or even purposeful. Think of someone who can only groan or laugh.
You can also sometimes see it to emphasize that someone communicating with sign language is both using language (thus, verbal) and is speaking (thus, speaking), but does not do so orally specifically. Other people would consider that non-speaking because signing is signing, speaking is oral speaking. And so on...
Mute is more complicated. At least in my view it's rather outdated when it comes to describing people who don't ever talk. The only places I still see it used is situational (also called selective) mutism, which is a real medical condition, and "deaf-mute", which is a pejorative term.
I don't really see many either non-verbal or non-speaking people also identify as mute, but out of the two it's probably closer to the "non-speaking" implication. But it generally strikes me as a term someone else would describe a person who can't talk, rather than for that person to describe themself with (with the exception of situational mutism). But it could be simply a matter of a social circle.
If any followers have additional insight feel free to share in the notes.
(addition by Mod Rock: deaf-mute is generally considered outdated terminology to refer to a Deaf person who doesn't speak, but there are some Deaf people who reclaim the term stylized as DeafMute to emphasize their decision not to vocalize as part of their Deaf identity.)