Also, AFAIK, the typical lack of integration among lines originally
intended
to compete with each other is even more pronounced in NYC than in London
where, as has been mentioned, the central tube lines were subsequently
extended into the suburbs along the rights-of-way (if not the tracks) of
the
'main-line' railways. , You will see far more stations idenified by the
'double-arrow' symbol indicating interchange with 'National Rail' on the
pocket route map for London than the corresponding symbols for interchange
with PATH, Metro-Rail and the LIRR on the NYC map. Of course, the local
topography may be relevant here.

This is oh so true. In NYC, there are actually very few stations that
interchange with the suburban railways, and only Penn Station has the
"national" railway running in and out of it, and that of course is Amtrak.
Amtrak does also run on the suburban tracks as you head out of the city. The
only stations that I am aware of that have the suburban railways
interchanged with the subway are Penn and Grand Central Stations, (Long
Island Railroad and MetroNorth, respectively.) WTC, (PATH) Jamaica Station
in Queens (Long Island Railroad), and Flatbush Avenue in Brooklyn. (Long
Island Railroad) There are other PATH stations in Manhattan, but can't
remember them off the top of my head.

Finally, after about a century of different fare policies (flat fare in
NYC,
distance-based fare stages in London), there now seems to be a convergence
towards a more sophisticated zone-based system with stored-fare cards...

Still flat fare in NYC, though they use the MetroCard now as opposed to
subway tokens. (thank goodness!) MetroNorth and LIRR have always had zone
based systems. PATH as I recall was always a flat fare system, though I
rarely used it.