Unless there is a compelling reason to remove the show loads like the real potential of having a branch break under this increased load, let the wind and moderating temperatures work and allow the branches to return to normal on their own. Tree limbs and evergreen branches are flexible and usually manage.
Trying to remove a heavy snow will likely cause more damage than left alone for the snow to melt away naturally. This is evidenced by all the needles mixed in with the snow after attempting to clear the snow off the evergreen limbs.
If there has been some damage, there is no real rush to remove the broken branch and cleanup the wound. Waiting until better weather is really ok. Use heading back cuts to repair the damaged limbs to allow the tree to regrow normally. Pruning back to a smaller branch will let the tree recover and grow out naturally after the repair. Fast growing trees or trees with poor structured canopies are likely to see more damage. Winter is a great time to take a look in canopies of your trees and look for structural problems like narrow crotch angles and crossing limbs. These situations may not cause immediate issues while the trees are young, yet set up conditions for later damage in stormy weather. Trees that have trunk damage early on allowing interior trunk decay will also potentially be a problem as they age. Not that a storm cannot take down a healthy tree, but what you see on the TV news after a big storm are trees that have rot up in the canopy, trunk rot near the soil line or those narrow crotch angles that have limited structural strength. Parkway trees are also likely victims as their root systems have been limited in size. Limbs broken at the trunk pose a larger repair problem compared to just having a branch broken in the canopy. As is the case in every situation, there will be those exceptions to the general rule. Limbs for example, hanging and presenting a health risk should be carefully dealt with of course. It will always be better to have an arborist who is trained to deal with these situations. They have the right equipment and protective gear to safely deal with removing damaged limbs. Homeowners should never attempt to clear power lines however tempting it may be.
There is a plus side to all this snow. Smaller landscape plants and perennials that are completely covered are not impacted by the really cold air temperatures.