Looking at the JFK weather last evening, the visibility at 6PM was about 1SM. Throughout the evening the weather was poor with low ceilings and visibility. A VOR/DME approach is a less precise approach than, say, an ILS approach, which requires the pilot(s) to see the runway from higher up and farther out than an ILS approach. Considering the weather conditions at the time, it may have been very difficult.
As to why they were doing the VOR/DME approach and not the ILS approach, I don't know. Part of the ILS may have been out of service for part of last night.