Irreligious (Dictionary.com 1st definition):
not religious; not practicing a religion and feeling no religious impulses or emotions.
Non-religious (Google definition):
not relating to or believing in a religion.
I've looked at definitions and synonyms for both, but I can't seem to find the difference between the two.
It appears "irreligious" has some kind of negative connotation directed at religion in general, while "non-religious" skews more to the "indifferent" definition.
However, I can't find any reliable sources that differentiates the two.