why did Nick want to leave Billy in the end

I don't think that Billy would really bond and accept being with his dad because Nick took on the father role that he needed. So while Nick was with them, then Billy would never really accept his dad.

A man can change his stars

His job was done.. he's a vagabond anyway, so why would he want to stay in one place, which he would have to do looking after the kid.

Obviously because Nick thought Billy would be better off with a man who deserted his buddy during a firefight, left his wife and kid to run off to Reno, gambled himself deeply into debt to the mob while cheating on his wife, and ended the film without a penny to his name but with the know-how to synthesize designer drugs.

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Lazy + smart = efficient.

Cuz that kid was a little prick, and Nick never forgave him for flipping him the bird near the beginning of the movie.
Oh, goodbye mama. I'll never do dishes again.