In the world of insurance, flooding and water damage are not the same thing. A flood is rising water moving over what's normally dry ground. Water damage is caused by water that hasn't touched the ground. If snow comes in through a broken window and melts on your computer, that's water damage, which homeowners insurance covers. If snow drifts in your garden melt, homeowners insurance won't pay for any flooding damage. Flood insurance will.
As long as snow stays off the ground, homeowners insurance covers damage from any melting. That includes ice dams blocking your gutters so the thaw flows into your house and snow blown into your house by a winter gale. Even collateral damage to snow that melts on the ground may be covered. If you leave your house because of flooding, for example, homeowners insurance would cover you if someone broke in and stole your stuff while you were gone.
Homeowners insurance can cover either the replacement cost of your loss or the actual cash value. When it's actual cash value, you get what your damaged property is worth. If you've had, say, your snow-ruined couch for 10 years, it's worth much less than it will cost you to buy a new couch. Replacement-value policies have higher premiums, but work better for you: You get enough money to replace the couch, even if that's more than the couch's cash value.
If your area get pounded by a major storm, call your insurer promptly if you have to file a claim. Chances are you're not the only one with damage, so you want to be at the front of the queue for getting a response. Take photographs that document the damage, and minimize further damage by covering leaks with plywood or tarpaulin. If the damage is minor, consider not filing a claim: Your company may drop your policy if it pays you more often than it considers profitable.