It is not noise abatement, it is airspace.
If JFK uses the VORDME to 22L (as opposed to the ILS), that allows LGA departures to depart RWY 13 and use the Whitestone Climb. Also, JFK landing RWY 13L (Canarsie) precludes the other LGA RWY 13 climb outs -- Maspeth and Coney.
The 'VOR' approach to 22L is really moot when it come to IFR/Visual approaches when 13L is also in use -- that operation is only allowed during decent VFR conditions (5SM and 2500CIG IIRC).
Also, the TNNIS RNAV SID off RWY 13 at LGA allows both ILS 22R and 22L to be in use at JFK, yet I believe it's still old SOP to use conventional nav SIDs off LGA when JFK is using 22L and 13L simultaneously.
No expert on the matter, but did visit both NY TRACON and JFK TWR.