A lot of comments have given their definitions of chaotic and neutral, but is no one going to address the "crazy"? This just shows a pretty poor understanding of mental illnesses and how they work.
Characters of any alignment can be mentally ill. There's nothing precluding mental illness even in a Lawful Good character. Mental illness doesn't make a person behave "randomly" or "just do whatever they want." Delusions or hallucinations may cause a person to behave based on information disconnected from reality, and some illnesses may also cause intense emotions, for example a person with paranoid delusions may believe that everyone is plotting to betray them and feel very afraid, but they're still reacting to these things as THEM. There is no mental illness that turns an average person into the Joker. Mental illness as an excuse for "random," irrational behavior, especially in villains, is a weak crutch for someone who has made their hero such an untouchable self-insert that no one could have a rational reason for disagreeing with them, and it perpetuates harmful stereotypes that hurt real people who suffer from mental illnesses.
If you want to write mental illness into your campaign, by all means research specific conditions, read up on the ways they present, and read accounts from people living with them. If your characters face or have faced trauma, read about how trauma can affect people. The human psyche can be a fragile thing, but it's incredibly lazy to just go to "ax crazy" as the default mental illness. When this happens as the result of trauma, it also warps things so that the REAL villain is always the victim, and shifts fear from the victimizer to the victim's justified wrath, painting that wrath as irrational. Ugh that trope is just terrible all over.
You're basically asking what the difference between mental illness and chaotic neutral are since they're both "ax crazy." The answer is that NONE OF YOUR CHARACTERS SHOULD BE AX CRAZY. As noted in the comments above, this isn't fun for anyone else, it's a stupid power fantasy for the player to play with being ~baaad~ and not having to suffer consequences, and it ruins everyone else's fun. Mental illness is a heavy and nuanced topic that should be approached with respect, and if the player isn't drawing from personal experience, lots of research. And chaotic neutral characters are not any more likely to be mentally ill than any other character.
I have a character who's CN, and not remotely mentally ill. She's basically someone who realizes she was handed disadvantages in life, and if she wants to get ahead in the world, she can't play by the rules, because the rules were designed to keep her on the bottom. She has a heart, and acts with empathy sometimes, but she realizes it's not her job to save the whole world, and that compassion disappears real fast if someone gets in her way. She recognizes that money is power, so she will do morally and legally dubious things to make money, though she does have some limits, and won't do anything for profit. (Drug smuggling, yes, human trafficking, no thanks.) She will, however, go along with the law sometimes in the interests of not getting in trouble--while she doesn't think the law serves anything but itself, and ascribes no moral value to it, she respects that it's powerful, and doesn't want to get skewered for her principles. She extends her selfishness to acting in the interests of those she likes and feels close to, but has the somewhat flawed bootstrap philosophy that if other people don't want to be victims, they should just be strong like her. People she hasn't accepted as close friends or family she thinks of more in terms of their usefulness to her. She's highly resourceful and adaptable, though because she thinks of herself as someone strong who doesn't lose, sometimes she has a hard time admitting when she's beaten, and will come back again and again with different tactics. Once she's decided to help someone, failing to do that would be a blow to her ego.
The difference between her and a chaotic good character, I think, is that she doesn't go out of her way looking for ways to make the world a better place, and she resists self-sacrifice. The difference between her and a chaotic evil character is that she isn't a sadist, and doesn't need to hurt anyone to feel good about herself. Her reaction to the bar selling tainted beer would probably be to not drink beer there, and tell anyone she liked to also not drink beer there--she feels no particular drive to put herself at risk resolving a problem that has nothing to do with her. She has enough of her own problems to deal with, without making every little problem in the world her personal cause. She's not a magic-user, but with the squirrel thing...she has no problem with hunting for food, but if she's going to kill something, there needs to be some kind of benefit to her other than sadistic joy at watching it writhe and die (since she's not a sadist). She doesn't think much of authority figures, but hardly sees the point in picking the people with the most power to hurt her and taunting them! She prefers to subversively get away with stuff they wouldn't like right under their noses, while smiling and seeming like a good citizen.
Her philosophy, boiled down, is less "greed is good" and more, "if I don't put myself first, no one else will and I'll never have anything good." She's decided to get the most out of life, and is willing to do whatever it takes. She's cunning and street-smart, but not random, ax-crazy, or soulless. She won't go running off to help strangers at the first word of a misdeed, but she also won't actively contribute to the senseless suffering of innocents. It's not that she doesn't care about people in need, she just doesn't see why it's her job in particular to fix it, and feels no responsibility to others, nor does she really see others as beholden to her. (With the exception of her family, they've always had her back, and she looks after them too.) She's someone who's broken the social contract because she felt it was holding her back more than helping her.
I don't completely agree with her philosophy--this isn't me using her as a mouthpiece, this is just me picking ideas that a person could have and could make a good story--but I don't think she's irrational.