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The Jaccard index divides the size of the intersection by the size of the union, so it penalises elements present in only one set. It is the standard measure for shingled documents, MinHash deduplication and tag-set comparison. The overlap coefficient normalises by the smaller set instead, which is more forgiving when one set is much larger than the other.
Jaccard index
J = |A and B| / |A or B| = intersection / (|A| + |B| - intersection); distance = 1 - J
Because the union is the size of the larger set. A 10-element subset of a 100-element set scores 0.1 on Jaccard but 1.0 on the overlap coefficient.
MinHash estimates the Jaccard index from a small number of hash signatures, so near-duplicate detection can run over billions of documents without computing exact intersections.