I'm traveling to NYC via Amtrak. There is a snow storm coming. Does snow cause trains to cancel their runs?

Snow can, but it has to be a pretty huge storm. You are probably pretty safe. The trains are more reliable than planes.

snow can delay a train, mostly through power outages that affect signals or perhaps cause the switches on the track to be difficult
the snow will not affect the train directly unless it is very very bad.
it will not cause the train to cancel but it may be late.

You are much safer on a train than on the highways, even if you have to wait awhile for it to arrive, you will get to your destination safely.

No, train service will not be canceled. The worst thing a snow storm will cause will be power outage to the signals along a section of the line, which will make train run late to New York. Otherwise, you'll be okay on a train than on a plane.

no usuaally can slow if icey and longer dwell times at stations but you should be able to get to your destination

Trains can travel through just about anything. Tracks are checked daily. You might be late, but you will get there. Just relax and enjoy the ride. Just be glad you aren't sitting on a tarmac in a plane and you can't get off. Or stuck in an airport and not going anywhere.

AMTRAK will get you there. I do recall about 1948 the Union Pacific's "City of San Francisco" got snowbound going through Donner Pass in the Sierra's and was stuck in the snow for 3 days.

I travel between Boston, NY and DC a lot. When the weather is bad the train is more reliable than the plane or dealing with traffic.