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

Because of blue man group

M. Night Shyamalan destroyed the public standing of the Avatar

sometimes i still say avatar and i wonder if it clocks me as a boomer (millennial)

when we sniff SEVERED our connection to a DIVINE FATHER

Fac*book

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

Pfp abbreviates easier

remember when the vernacular “private messages” mysteriously got replaced with “direct messages” across nearly all tech websites?

‘avatar’ calls to mind customized 3d character models like in an MMO videogame or nintendo’s ‘mii’ feature, personally.