While something not conforming to the established rule, method or usage and which is out of the ordinary is termed “irregular”, anything which is prohibited by law, against the law, unlawful, illicit and not authorised or sanctioned as by rules is termed “illegal”.
ACIT v. Vijay Television Private Ltd. (2018) 407 ITR 642/ 304 CTR 149/ 169 DTR 17 (Mad) (HC)
Words and Phrases- Meanings of -“irregularity” and “illegality”.