IIRC, minhashing is used to approximate Jacquard similarity (a set-theoretic measure), while random hyperplanes (aka simhashing) is used to approximate cosine s

IIRC, minhashing is used to approximate Jacquard similarity (a set-theoretic measure), while random hyperplanes (aka simhashing) is used to approximate cosine similarity (a geometric/algebraic measure). So they solve different problems, even though some problems can be cast in terms of either framework.