I feel like both have been used since forever. For example here's a Newgrounds thread from 2005 where everyone uses the term 'profile picture'. Here's the term 'avatar' being used in Neopets in 2003.
To trump both, here's a forum post in 2001 using both terms
Avatar is for websites where most people prefer to be anonimous, pfp is for the opposite
And since everyone here is a fed, it tracks that we say pfp.
Because of blue man group 
M. Night Shyamalan destroyed the public standing of the Avatar
It's called Avatar because a lot of the shots are close ups on the face
sometimes i still say avatar and i wonder if it clocks me as a boomer (millennial)
Another good one is folder vs. directory.
"directory" can expose you as a unix nerd as well
App(lication)/program too
Did we ever call them avis or am I misremembering that
We definitely did. I still use that term from time to time.
when we sniff SEVERED our connection to a DIVINE FATHER
Fac*book
Fa**book = Fartbook?
pfp? pig foop palls
When we started having profiles instead of user pages, and we started having pictures of ourselves instead of anynomous representations of us.
The meanings are slightly different.
An avatar is a body that contains the soul/mind of another being, like a deity. Could also be a human being implanting their mind into a blue alien like in the movies. Or it could be Aang’s body being the physical vessel for ancestral benders of many tribes.
On computers, an avatar is the actual character that you build for a game or website. For example, Yahoo used to have avatar builders. Then features were added where forums would show a picture of your avatar and people would call the picture itself the avatar, as shorthand. But they’re two distinct things. A profile picture does not need to be a representation of the person (avatar), it can be a picture of anything at all.
handle
I still say Avatar, but then again most of my habits are rooted in 20 years old ‘net culture. I still say “Emoticons” from time to time
Same, also "image macros"
Pfp abbreviates easier
Avi
remember when the vernacular "private messages" mysteriously got replaced with "direct messages" across nearly all tech websites?
Because they aren’t private anymore 
Pfp looks like typing a wet fart.
Or it's the first letter in the secret Duck Alphabet they use to chronicle the sins of man.
'avatar' calls to mind customized 3d character models like in an MMO videogame or nintendo's 'mii' feature, personally.
I think it was originally that, then default, then avatar, then back to that
Avatar dates to early graphical bulletin boards and MUDs in the 80s. Originally it was used as a synonym for "Player Character" (after Moorcock and his Avatars of the Eternal Champion) and spread from there. PFP is a post-facebook thing.
I wonder if the Ultima series had anything do with that, since the player character was called the Avatar. NATOpedia seems to support that, claiming that Ultima IV popularized the use of the term avatar in this context.
Yeah, I think it was Moorcock > Ultima > MUDs > BBS > General adoption
I don't think so, usage of avatar predates most social media, pfp was popularised i think by facebook boomers and purest corpo shit like linkedin.
Yeah that sounds right
"Avatar" sounds too much like a movie title or something.
PPB, profile picture balls
retvrn to phpbb
This but unironically.
Believe me, I was not being ironic
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