3 year MD-PhD student here.
From the perspective of MD students:
Weill Cornell and HSS labs are essentially the same thing, many clinical and translational/basic science opportunities
MSKCC labs are very well connected to medical school, (clinical/translational/basic research and rotation wise). There might be some administrative quirks (e.g. you need to get separate id), but usually easily solvable.
Rockefeller University - mainly basic science research labs with some translational ones, you can do research here if you reach out to lab heads directly (each lab is its own department), but rarely done since basic science usually requires serious time investment (not very suitable for AOC, but doable if you reach out yearly in M1/M2 and do research throughout school, but then why not do MD-PhD instead?). Less connected to medical school (however MD-PhD and PTSP residents often do their research here)
Cornell Tech - some collaborations are brewing, unexplored for the most part
Cornell Ithaca - distant relative, no serious connection except the name and some initiatives by administration.