While Planescape and Forgotten Realms are both D&D caimpaign settings, I am pretty sure they are seperate settings. Planescape was released as a setting in 1994, while Forgotten Realms was released in 1987.
http://en.wikipedia.org/wiki/Forgotten_Realms
http://en.wikipedia.org/wiki/Planescape
I am likely wrong, since I have only played D&D via video games, but I remembered reading that the two realms were different. Reading the Wiki seems to confirm that to me, but then I only had time to skim it before posting.
You're right in technicality. Planescape represents a host of different planes.
Outer Planes - Places like the Abyss, Hades, Pandemonium, Elysium, Ysgard, and so on. Originally there was no known number, but in the later editions they made it so each plane was matched with an alignment (Chaotic Good, Lawful Evil, Neutral Good, etc). This is where most of the gods were found and where the souls of dead people from the Material planes would end up.
Inner Planes - Planes based on the elements. Fire, Water, Earth, Air, Negative Energy, Positive Energy. Pretty inhospitable places as they're made up entirely of that element.
Material Plane - Worlds like Faerun (Forgotten Realms), Krynn (Dragonlance), Athas (Dark Sun), etc exist all in the Prime Material plane. Another D&D system, Spelljammer, was based around traveling between these worlds through the Phlogiston.
There was also the Astral and Ethereal planes which were basically the links between the Material and Outer/Inner planes.
Planescape Torment takes place mostly in Sigil, the City of Doors. It's located smack dab in the middle of the Outer Planes and has portals all over the place to every plane.