You pay all costs and follow all normal timing rules for spells cast this way.

Shadow of the Enemy uses a new template indicating that you may spend mana as though it were mana of any type to cast the exiled cards. The six types of mana are white, blue, black, red, green, and colorless.

Snow mana is not a type of mana. Shadow of the Enemy won't let you pay a snow cost using mana produced by a nonsnow source.