I don't know about New York in particular, but Chicago water engineering seems a related topic.

Here you do deep tunnels to avoid the surface, in ways another poster said; everything is easier when nothing is in the way.

For the mathematical difference, 400 feet below sea level and 800 feet below are almost exactly the same: difficulties are water getting in to your pit, but the machines that work on rock, work on rock at the same speed regardless of depth, so the difference between 400 feet and 800 feet is best described as 400 feet difference. A big issue here is that they do not drill; they hammer. Pounding base pylons into bedrock causes dramatic rhythms in the surrounding 500m, but that's to deal with the bedrock, not depth.