Or it can be colder and snowier, we had several instances of this during the 1980s. Vet Day 1987 is an example of a storm when it snowed in DC and rained in NYC. This used to be much more common back then.
There was a storm like this in December 1989 a very cold month, when DC got snow and NYC flipped to rain right at the onset of the storm. A huge busted forecast of 6-8 inches of snow ended up being rain and thunder here.