Why must we distinguish between rational and irrational numbers? The difference between rational and irrational numbers is always stated as: rational numbers can be written as the ratio of two integers, and irrational numbers can't. However, why do mathematicians make a distinction between these two types of numbers? Why are integers special anyway, other than being historically significant?
Is there any property that sets rational or irrational numbers apart, other than the way they are written in our number system?

Another example where they differ. Take any polynomial $a_nx^n + a_{n-1}x^{n-1} + \ldots +a_1x + a_0$ with integer coefficients.
You can easily find all rational roots of that polynomial: Any rational root $\frac{p}{q}$ (with $p$, $q$ relativly prime integers) must satisfy: $p$ divides $a_0$ and $q$ divides $a_n$. So there are finitely many possibilities which you can check by hand.
It's not easy in general to find irrational roots. 
