Quick answer
Both are fire-performance sheaths, but they are not the same claim. LSH (low smoke and halogen) means reduced smoke and reduced halogen content — the material may still be PVC-based. LSZH (low smoke zero halogen, also written LSOH or FR-LSZH) means the compound contains no halogens at all, so a fire produces no corrosive halogen acid gas. Zero-halogen grades are the stricter choice for enclosed public spaces and equipment rooms. On house wire the same split appears as FR-LSH (PVC-based) versus HFFR (halogen-free) — our guide to choosing the wire grade covers it.