ABET accreditation is specifically important for those who want to become licensed as a Professional Engineer (PE), which is mostly important in civil engineering. It is also specifically important for those who want to take the patent exam.
It is not specifically important in many other contexts, but is helpful in indicating a relatively high minimum standard of quality for an engineering degree (particularly in one of the more traditional engineering fields like mechanical or electrical) at a lesser known college. Some colleges with strong reputations in engineering no longer go through ABET accreditation for some of their engineering majors because it is extra work for them (but they continue to do so for civil engineering).