Why do planes make landing approaches at significantly varying heights?
Unfortunately, since I don't know your particular location relative to the airport, I can only give some general answers as to why this might be.

Contrary to popular belief, the availability of precision approaches does not mean they are used 100% of the time, nor are aircraft always routed the same way. Depending on the airport, the use of precision approaches can drastically reduce the number of aircraft that can be handled in a given period of time.  As such, during visual meteorological conditions, flights will almost always be cleared for a visual approach. Busy airports will typically have a number of standard arrival routes (STARs) that make the approach routing consistent and predictable (which could explain why noise is not an issue 99% of the time), but pilots are not obligated to use them.  If a STAR is not used during a visual approach to an airport with approach control services (like JFK):


The approach controller (using radar) will direct the plane in such a way to allow the pilot to visually acquire the runway, at which point the controller will clear the pilot for a visual approach. Additionally, under radar vectoring, the aircraft may be told to fly a particular heading for sequencing and/or spacing which can vary due to the amount and type of traffic. 
Once cleared for the visual approach, the particular approach course and altitude is at the pilot's discretion.


The situation is similar for an instrument approach, except that the approach controller will arrange to have the aircraft either navigate to an initial approach fix on a published approach procedure or vectored to intercept the localizer before clearing the pilot for the approach.

Sometimes, unusual flight paths can also occur when a runway that is not normally used for landings is made active.  This can happen when wind favors a particular runway, or the one that is normally used is taken out of service (e.g. maintenance or an accident).