Why wouldn't KJFK offer the ILS for 22L? (Source: Flightradar24)
The above was from last Sunday around 17z. I was watching FR24 with the ATC feed. The airport was operating in 'arrival priority' with approaches to both 13L (Canarsie) and 22L.
Both directions were offered their respective VOR approach. Authorized carriers such as JetBlue and Alaska Airlines requested the RNAV RNP 13L (not publicly published by the FAA, yet).
My question is why wasn't the ILS offered for 22L? I checked the NOTAMs and the serviceability of the ILS was not there. And since the VOR is an instrument approach, it wouldn't quite fit the visual vs. instrument asked about here. A 4-year old post on airliners.net blames LaGuardia (LGA):

JFK will frequently land with max crosswind or tailwind components and frequently uses off set VOR approaches to minimums before switching to the preferred ILS.
Why? LaGuardia and the airspace to the north.

While yes the VOR is offset (232° vs. 224°), it's not a big offset that would free a chunk of airspace for LGA when it vectors its inbounds via a left downwind for runway 22. To be sure I superimposed both plates by aligning the 604' structure (shown below).
JFK ARR ILS 22L/22R DEP 22R


  If conditions dictate the use of JFK ILS RWY 22L approach, then (1) the JFK Area owns Belmont airspace 3,000' and below, (2) releases Coney airspace to LGA and LIB, (3) JFK cannot depart 31L, (4) ILS 22R unavailable if LGA departs RWY 13.


LGA ILS 22, DEP 13


  LGA Coney, Maspeth, and Whitestone climbs are available.


LGA ILS 22, DEP 13 (No Belmont)


  If conditions dictate the use of LGA ILS RWY 22 while departing RWY 13, then LGA departures are restricted to Maspeth and Coney climbs only*.


So, based on those two things, if LGA doesn't own the Belmont airspace, then Whitestone Climbs (WC) are disallowed. Therefore, when JFK uses VOR 22L, it allows LGA to use the Whitestone Climb off of runway 13, which is preferred for noise abatement.

*Personal note: this application was created prior to the addition of GLDMN, NTHNS, TNNIS RNAV SIDs which are now used instead of Maspeth and Coney.

Sources: 


Previously based in LGA at a regional airline
NY Airspace Application 
http://tfmlearning.faa.gov/NY_Airspace/NY_Airspace_Pkg/NY_Airspace.swf
(Only works in Firefox)