"North Shore/LIJ" is the name of the hospital system, the largest on Long Island. The two flagship hospitals of the system are North Shore University Hospital in Manhasset (academically affiliated with NYU) and Long Island Jewish Hospital in New Hyde Park (academically affiliated with AECOM).
Most of the training programs are separate, though some are merged or in the process of merging.
As to which is 'better' - they both have reputations for being fairly solid IM programs but not exceptional. When I was a student, North Shore IM tended to take American graduates only (MD and DO, and some Sackler students) while LIJ's medicine program only accepted US MDs and FMGs. I'd say that the programs are at least on par with the mid- to lower tier Manhattan programs academically. Important for some is that both hospitals have decent fellowship training opportunities.
In terms of location, they are only a half hour or so from Manhattan so you get the benefit of living close to the city without paying ridiculous amounts for rent.
