Wes Anderson’s 2001 film The Royal Tenenbaums features a short exchange that’s interesting for its taboo-linguistic detail. It takes place between Royal himself, played by Gene Hackman, and Henry, played by Danny Glover.
If you haven’t seen the film, but you might sometime (do, dammit), don’t worry about spoilers – the images below don’t give much away. And you don’t need to know the characters’ backstory, so let’s jump right in:
I love pretty much everything about this conversation, not least Royal’s poetic use of blue to mean ‘sad’ – though here on Strong Language it tends to have another denotation.
And with those insults, too, the nuances clearly matter. So what is the difference, would you say, between an asshole and a son of a bitch?
Strong Language 
To me the difference between asshole and son of a bitch is clear. An asshole does mean and vile things primarily for pleasure, while a son of a bitch does them of necessity (although that would not preclude the taking of pleasure from such an act).
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There’s something to that, though to me an asshole generally does mean and vile things because it suits them, and doesn’t much care how others may be affected (or may take pleasure in the fact that others are affected negatively). A son of a bitch, by contrast, may even regret that others are affected negatively, but this won’t stop them acting as they do.
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Another perspective via Twitter:
This is supported by the relevant entries in Green’s Dictionary of Slang, where asshole can mean ‘a fool, a derog. description of a subject’ (since 1933 – newer than I expected), while son of a bitch can be ‘a derog. general term of abuse’ (from 1613) but also ‘an affectionate term of address; also used self-reflexively’ (from 1847).
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