If the creature this Aura would enchant is an illegal target by the time Winter's Rest resolves, the entire spell doesn't resolve. It won't enter the battlefield, so its ability won't trigger.

If Winter's Rest is destroyed while its enters-the-battlefield triggered ability is on the stack, the creature that it enchanted is tapped as that ability resolves.

If you lose control of all of your other snow permanents, then control a new one, the effect of Winter's Rest will apply to the enchanted creature again.