I know they all refer to unusual behaviour in a person. But in modern life while you work and you meet a native speaker of English in America and then there is a conversation what do the words really mean?
From American Heritage Dictionary -
Nut (slang) - a crazy or an eccentric person
This would indicate that the terms nutty, crazy and eccentric may be interchangeable in some contexts.
Now my opinion (sorry, not a native though!).
When you call someone 'crazy', it adds a bit of danger there (Don't go near him, he's crazy, he can beat up you); nut is more kind of a fool, funny looking OR (at times innocent? -I remember Mary Jane calling you are such a nut to Peter Parker); eccentric is a person with odd or unusual personality.
In order of severity, from greatest to least:
This can be very insulting and is only used
informally. Similar to: nutcase, nut job, loony, etc.
There is, however, a non-insulting use: It can also mean
very enthusuastic about somethinglike a hobby or interest as in sports nut. Meaning: a person who loves sports a lot.
Although less informal than the first one, the two terms are fairly the same. However, you may hear it used lightly to express surprise or disbelief like in this scenario:
Jesse: "I broke up with James last night."
Meowth: "That's crazy."
It can mean:
slightly strangeor just
weirdas in eccentric taste
unconventionalas in eccentric billionaire
Although, this may be used as a euphemism for the first two terms.