This may seem like a fatuous division. After all, what is the difference? Let us examine the consequences of who they are style thinking. For ease of use we can coin the terms existential (who they are) and agency-based (what they can do) just to make what we're talking about a little easier to digest and perhaps fill the conversation with complicated jargon to confuse outsiders. Anyhow! Existential thinking in character creation does not ask you, necessarily, to find rules reflections in who your character is. More important than his statistics and skills is his past and personality, and in older roleplaying games it should be a very sparse past indeed.
Using AD&D 2e as an example (an interesting example to be sure, as it rides the crux between existential and agency-based thinking and the system can be turned either way) we can see existential based thinking in the following scenario:
My friend, Tallstaff, rolls up a character. The first and foremost thing done when rolling a character by Method I is asking the dice for your stats. You can really do precious little planning unless you're allowed to move your stats around, and in this instance he is not. He rolls an 8, 17, 12, 9, 11, 15. Right off the bat we can see that this character is going to be a thief.
We already know he is probably going to get the standard thief abilities (unless he chooses to take a kit, which is already straying into agency-based character building) so his next job is to explain his stats. Why is he so nimble? How has his average intelligence hindered him in life? Have his good looks (or fun personality) made him cocky? And yet, these sorts of things are decided on a paper-thin level and can easily be overwritten; he is going to play this character for a level first to see how exactly his personality forms.*
So, the majority of his creation is determined by the flavor that is applied to him, rather than the rules he has access to. His personality is not framed by unique spells and powers but rather by things that happen to him in the course of his life. He is formed by experience; there are precious few "powers" for him to even have, save the standard thief abilities.
Now, we can use 4th Edition D&D to make the same sort of argument about agency-based thinking so we can examine that.
Since Tallstaff would never play 4e, I will just go through this character creation by my lonesome. Opening my books and getting re-aacquainted with the game logic, I recall that my first task is actually to decide what kind of character I want to play. According to the PHB the first things I need to pick are a race and a class. Because I don't want to make something too complicated (this is just a blog after all, I don't need to be busting my ass to make a combat-monster) I will make a human fighter.
The standard method of having scores is just to have 16, 14, 13, 12, 11, 10 arranged to taste. I'll keep 'em in that order. On a note unrelated to this experiment we can already see that characters in 4e have a flatter statline and are in general, assumed to be above average by a great deal.
Usually, in the older games, you don't "get" anything for being human. That's considered the baseline. In 4e, you receive a suite of bonuses (to counteract the bonuses other races get). Alright, that's not how I prefer it but it doesn't push 4e any closer to agency-based gaming, as there are races that get bonuses in all editions.
Now we're down to the nitty-gritty: power choices. Like feat choices in 3.x, these are the flavors that will describe who your character is. Unlike 3.x, all powers are combat based, so these flavors tend to be about how you fight. The difference between me and any other fighter at level one is going to be primarily understood in differences of power choices (feat choices in 3.x). This is what makes me unique. This is agency-based gaming: the difference between characters is mirrored in the rules by picking-and-choosing extra rules to apply to yourself.
So we are no longer different because of our stats or our pasts or our personalities (or at least, these things are not our primary differences) but rather because of our builds. It is the rules-reflection of our character which determines our conceptualization of him or vice versa. This reminds me of the main characters in pulpy young adult fantasy: they are always gifted with some magical ability that no one else in their setting seems to have. The last scion of an extinct race, the secret son of the king who can command the forces of nature, whatever it is: this is it.
It seems to me a shallow road indeed to differentiate ourselves from one another primarily by rules. After all, isn't there something similar to the way all people fight with a sword and shield? Perhaps some of them are slightly better at some aspects of it, perhaps some of them rely more on other maneuvers, but it is a strange world where being able to shatter a man's armor (brute strike) is considered mutually exclusive with being able to hound that man with skilled parries (villain's menace).
*This is one of the reasons why I really don't like allowing people to make higher level characters to begin with, because it robs them of the critical character-forming period when they're level one. The way you react to stuff when you are an average schmo really informs the way you will behave when you're level 30 and thinking about becoming a god.