What's the difference between "eldest" and "oldest"?

Indeed, both eldest and oldest refer to the greatest in age. The crucial difference, however, lies in the fact that eldest can only be used for related persons, while oldest can be used for any person, place or thing in a group of related or unrelated elements. Examples:

• He is the eldest/oldest of the three children.
• Mine is the eldest/oldest car on the block.
• John is the eldest (less common)/oldest student in my class.
• She is the eldest (less common)/oldest of my nieces.
• 'Is New York the eldest/oldest city in the US?'
• He's the eldest (less common)/oldest in the brotherhood.

And while eldest can be used for any group of related persons, in reality, it is mostly only used in reference to siblings.

New Oxford American Dictionary (2nd Edition) definition of eldest:

(of one out of a group of related or otherwise associated people) of the greatest age; oldest

They are interchangeable; they both mean exactly the same thing. I think there is a general tendency to use eldest in relation to people, e.g.

John is my eldest son.

For some reason,

John is my oldest son

sounds wrong, almost like he's the son you've had for longest as opposed to him being the son with the greatest age.