When you say that someone is ‘immoral’ what you mean is that the person does a lot of things that are not morally acceptable; he does things that are positively wrong. When the word is used with people, it is always used to show disapproval. Perhaps the individual swindles people, has extramarital affairs, etc.
Example: What you are doing would be considered to be immoral by most people. *
A person who is ‘amoral’ does not know the difference between right and wrong, and he doesn’t really care whether what he does is morally right or wrong. He is not concerned with morals; he is outside the sphere of morality. Unlike the word ‘immoral’, ‘amoral’ is not always used to show disapproval.
Example: According to the critic, the hero had an amoral attitude towards murder.
Source: “The Hindu” – Know Your English Column – Jan 22, 2008