There was no sudden warm up, different time, different climate. Snow actually stuck around in NYC back then, not like the 2000's, big storms (except for the last two years) and mild winters.
The month of March 1888 the average temperature in NYC was 29.9° for the entire month. Consider that against the current January 30 year average in NYC is 32.6°.
Also consider that it's been over two years since NYC had a month that averaged below 32° which was January 2018 at 31.7° and before that you'd have to go all the way back to February 2015 23.9°. Several high impact snow storms mask the truth, but for NYC it's been a horrible stretch of above normal and way above normal winter temperatures.