Functionally, not much I can see , the former's slur for people with NPD (or whom person saying it thinks has NPD)

Narcissism, like all personality disorders, is a trauma response and also not indicative of someone's moral character as you can get help and learn not to act on your worst inclinations.

Asshole is a behavior.

Narcissists lack something that makes them whole. They desperately try to fill that void that fits in the current moment. They tend to truly believe a story to fit their needs and completely change it when the situation changes. You won't be able to correct their story based on your own memory. Sociopaths are whole, their memory is stable, their deeds are purposefully malignant.

There's gotta be a vin diagram for this one

My layman’s understanding is that clinical narcissism is a pathological obsession with the perception of self (vs. sociopathy which is pathological obsession with self-interests, explained in a moment), and being an asshole is a transient state.

A narcissist will constantly have in mind what others think of them, and obsessively make sure other people believe they are good, socially better, morally superior, high status, etc. at any cost, and will topple and tear down other people to ensure that perception. Anyone who challenges that is a threat and will be treated with active hostility. They care what you think because it feeds their sickness.

A sociopath generally does not care what you think, unless you believing something is useful to them. They are typically equally focused on others’ perception of them but for a different reason. People are simply tools; means to an end. Reputation is an asset in their toolkit to get what they want, at whatever cost. Lack of remorse at a pervasive, chronic level tends to characterize sociopathy.

An asshole doesn’t necessarily mean either. I can be mean to you, or hurt someone to get my way, and that would make me an asshole. I don’t necessarily think I’m better than you and I do have the capacity to feel remorse. Asshole is a changeable behavior, or an attitude, and not a severe disorder.

TL;DR: Narcissists are specifically mentally sick, and can be assholes. But anyone can act like an asshole ¯_(ツ)_/¯

Every narcissist is an asshole, but not every asshole is a narcissist

The asshole is an asshole on purpose.
The narcissist couldn't stop acting like an asshole even if they tried.

a narcissist is self-absorbed

an asshole actively enjoys the suffering of others

One of them narrowly made it into the DSM-5 and the other never will.

The Politics of Sociopathic Narcissism

It is thus plausible to consider “narcissism” and “sociopathy” as really two poles on a continuum of overlapping traits (and indeed, the newly-revised DSM V, which considered subsuming narcissistic traits into sociopathy, has moved in this direction).

Youtuber Sarah Z has a whole video about how this online "Narcissist" obsession is dumb.

Narcissist believes they're right. Assholes just think you're wrong.

Depression

anyone performing actions that hurt others where its not really about themselves. being mean for no particular reason or reward, especially to themselves.

NPD and BPD are both driven by an eternal sucking void of insecurity and negative self-esteem that can never be filled.

Narcissists try to fill it with praise and social status, or failing that, power and subjugation.

Borderlines try to fill it with limerence and victimhood or failing that, revenge.

But it can't ever be filled - it's worse than addiction, worse than being underwater in some shitty MLM scheme. It takes and takes and takes because fuck you, pay me.

So they don't have friends or loved ones, they have hosts. They suck out all the goodness until there's nothing left to give, then they start in on the pain.

Assholes, in the other hand, are just driven by simple greed, callousness and lack of consideration. They can and do still hurt people of course, but it's not an all-consuming obsessive need.

A formal diagnosis. I know narcissistic is thrown around quite a bit but I prefer to use the term self-centered when not talking about NPD as I have a relative with NPD as one of many comorbid disorders.