As sets the integers and rational numbers are identical. They can be put in one-to-one correspondence with each other.
To distinguish them further we must impose additional structure and examine their differences.
Algebra is the study of operators over sets. This means from an algebraic perspective the difference between the rational numbers and the integers are the operators supported. The integers support addition, subtraction, and multiplication. The rationals are formed by adding division.
On the other hand, topology is the study of how to formalize the intuitive notions of "close" and "far." From a topological perspective, the distinguishing feature of the integers is that each integer is very far from the other, or equivalently that the integers are very sparse. Every integer has two other integers between which there are no other integers. On the other hand the rational numbers are very dense. They are the result of taking the integers and "squishing" them together. Between and two rational numbers there are an infinite number of other rational numbers.
The set of rationals, Q, is homeomorphic to their Cartesian product, Q x Q. So any analogy needs to work for that set, too.
In particular I was thinking of undistinguished countable sets (although I was confusingly throwing in ordering of the integers to try to make my point more accessible) that you then add topological structure on top of.
In that world the integers are simply the discrete topology on a countable set. Or more explicitly (to contrast with the next definition), where all singleton sets are open.
The rationals then are formed by any metric whose induced topology does not include singleton sets.
That is, any attempt to uniformly bring elements "closer" than the world where single points are open gives rise to a topology homeomorphic to the rational numbers.
