"Violent crime" is not a consistently defined or tracked category across cities, so it isn't comparable. The NYPD has a notoriously... loose definition of "violent crime", to the point where it counts things that no reasonable person would be thinking of when they hear that term.

For this reason, researchers typically use homocides to make comparisons, because that's consistently defined and tracked across jurisdictions, and because it's harder to manipulate those statistics when recording.

NYC - particularly Manhattan - has a much lower homocide rate than other places.