I think irrational numbers are for specifying an amount of something accurately(or far more specific if not accurate) up to infinitely small scale and can never be reached by getting the ratio of any 2 whole numbers(integers). Examples: amount of liquid, exact volume of a container, exact length of a rope around a drum, density of moisture in the air. (Though, when computing these quantity, we get rational numbers. But that can't be easily know.)
Rational numbers occur when counting some whole/countable objects like apple, orange, houses, trees, ballpens, etc, are involve, which getting the count of an object as a whole(ignoring their sizes) are more important than getting their volume, or the space they occupy nor their weight.
Example: "how many coins do you have right now?" ask for a rational number while "how much profit you earned in your bank last year" ask for a quantity that has a whole number count and a an amount less than a whole number to specify exactness.
Non-integer rational numbers still involves whole numbers. Example: I have apples double your number of apples, or your apples are just 50% of my apples. We know that 50% comes from 50/100 or 1/2. If we know these, 50% or 0.5 is obviously a rational number. They are accurate if expressed as fractions compared to floating point notation which involves rounding off when writing them to save space.
I think, rational is more on finite measurement while irrational is more on infinite measurement(infinitely small approximations). I'm just answering based on my opinion and have no complete info about these two set of numbers. But for informal explanation that is my answer. I hope it helps.