I'm of the opinion that there a few things that largely help make Planescape work, in my mind, and are likely to feature in any PS game I run.
The factions are one aspect, and allow the D&D alignment model to show ways that two people with the same alignment can think VERY differently and largely disagree about most things. A LN Harmonium member probably thinks very differently than a LN Sodkiller, while a CG Sensate and a LN Sensate might prove to think quite similarly, with the comparatively minor exceptions forming the basic difference between their alignments. Without the kriegstanz and the faction-based classes, even if it's just in the background, there is not quite the same ability to show that belief really IS power.
Likewise, another aspect I think that one pretty much needs to have is some concept of planar politics. Not just between factions, but also between the various groups of outsiders, and how it can be that a LG Archon can occasionally work quite happily with a LE Devil (and vice versa...) when there is a benefit for doing so, like there might be if they are attempting to take down the same CE Demon. Most of the time, you won't see that going on much in a D&D campaign world; most people play up Good vs. Evil, and many leave Law vs. Chaos in the dust. PS blows that assumption flat out. There are fiends with holy weapons BECAUSE it makes sense for their 'allies' on the celestial side to give them those weapons so they can take out their common enemy, and if the fiends use them on members of their own 'race' it's fine and dandy as well.
I also like the idea that the "Prime Material" both matters quite a bit and very little at all on the Planes useful. The Prime Material matters quite a bit because the Primes are a source of innovation and house an incredible store of belief to be tapped; the Prime material matters hardly at all because a given Prime's influences are largely limited to small planets, while the Planars can have contacts and influences are spread across multiple infinite planes that even encompass the territory the Prime knows.
Psion: you may start in Curst, the City of Glass, or some other Planar city, but you also do so in the recognition of Sigil. The planar metropoli are important because of connections. Sigil is a city of connections, as it is essentially the center of 'neutral' territory, and home to innumerable portals known and unknown.
Things I don't find as important: the planes themselves (shock, shock!) and the level of magic/technology that's implied with the Planes currently. If I were to run a modified planescape, I would probably change things to take things off the Great Wheel. I would keep the Greet Wheel planes, and make them ones the PCs are likely to come across, but I might consider adding a large number of other 'smaller' planes to show the diversity of life you get when you and a bunch of other people can pick a favorite spot anywhere in the whole Multiverse to put up kip. Likewise, I am perfectly OK with updating planescape to the modern era, or dropping down the magic/tech level a bit.
