Describes a person who is sharp, misses no opportunity, and is often business-savvy or socially perceptive. Can carry a slightly negative nuance of being calculating.
鋭い means sharp in a broader sense (keen senses, sharp mind), while 抜け目ない specifically implies not missing practical opportunities.
Etymology
From 抜け目 (a gap, oversight) + ない (not), literally 'without oversight'. The exact historical derivation is uncertain, but the phrase has been used idiomatically since at least the Edo period.
