The difference between Irrational and Nonrational
When used as adjectives, irrational means not rational, whereas nonrational means contrary to reason.
Irrational is also noun with the meaning: a real number that can not be expressed as the quotient of two integers, an irrational number.
check bellow for the other definitions of Irrational and Nonrational

Irrational as an adjective:
Not rational; unfounded or nonsensical.
Examples:
"an irrational decision"

Irrational as an adjective (mathematics, arithmetic, number theory, not comparable):
Of a real number, that cannot be written as the ratio of two integers.
Examples:
"The number π is irrational."

Irrational as a noun:
A real number that can not be expressed as the quotient of two integers, an irrational number.

Nonrational as an adjective:
Contrary to reason; lacking an appropriate or sufficient reason; irrational.

Nonrational as an adjective:
Lacking the ability to reason.

Nonrational as an adjective (often, philosophy):
Not within the domain of what can be understood or analyzed by reason; outside the competence of the rules of reason.

Nonrational as an adjective (economics, social sciences, public policy):
Not based on one's own interests; inconsistent with utility maximization.