It's not even a debate, Ellie is an infinitely superior character. Purity doesn't make Elizabeth a better character, it actually does the opposite in some respects. She's the princess in the tower, the girl who is too perfect for the world. She almost borders on mary sue, though there's enough characterization that she becomes more interesting later on. Elizabeth felt a little cliche to me, and I felt the game missed the little intimate moments that made Ellie great. She's also a very reactionary character who only responds to the things happening around her. Her future is completely shaped by the people in her life, until perhaps the very end. She's a good character from a good game, but she's certainly not great.
Ellie, on the other hand, is flawed in just the right way. She can be naive, but tough in a way Elizabeth isn't. She's also resourceful, and takes a situation into her own hands before waiting on another character to rescue her. She manages to be vulnerable without being weak, something that is VERY hard to pull off in writing. Elizabeth rarely succeeds at this. Ellie is funny, brave, clever, a little bit pervy, and very very flawed. In other words, she's a human being. More specifically, she's a teenager. It's been said that we like people for their strengths, but we love them for their flaws, and in this Ellie feels like a well rounded and more lively person than Elizabeth ever was.
I would also like to point out that Ellie felt more like a real, fleshed out character. I wanted to protect her as Joel, but I also wanted to protect Joel as her. This is very important, because it solidifies her as an independent character independent of Joel. She's not the sidekick, or "the chick." She's just a normal person, and Naughty Dog never felt the need to trumpet the fact that they had a strong female character. They were confident in their writing, and let the characters speak for themselves. Elizabeth always felt like she was tagging along for the journey, even though she was the center of the story. The game tried too hard to make me feel like I needed to protect this woman. The Last of Us, on the other hand, felt like it was simply a story of two characters looking out for one another, and both of them were needed to keep the other alive.
That said, I hesitate to label anyone "the greatest of all time." Even someone as lively as Ellie.