“no hay ”es un verbo,puede formar un frase.Pero "sin" no puede componer un frase.

The Literal translation is 'No hay' means 'There is no' and 'sin' means with out. if you say 'no hay', it means, that there used to be something but that is no longer true. i.e : No hay mas leche (there is no more milk) Habia leche, pero no hay más. (there used to be milk, but no more). Sin mens without, you could understud this by examples, here are some se marchó sin ellos, he left without them; costó mil pesos, sin contar el IVA, it cost one thousand pesos, not including VAT; el edificio estaba sin terminar, the building was unfinished; entre sin llamar, come in without knocking; saldré sin que me vea, I'll go out without him seeing; una bebida sin alcohol, a non-alcoholic drink.