Joe Siegler's explanation of the relationship between Apogee and id follows:

Apogee and id Software are two completely different companies.  They
always have been, always will be.  Apogee doesn't own id; id doesn't own
Apogee; nor is one a division of the other.

Apogee Software writes and distributes software.  Of the many games that
Apogee has published, only a handful have not been written by Apogee as
well.  id Software wrote three of these games, namely, Keen: Vorticons,
Keen: Galaxy, and Wolfenstein 3D.

id Software writes software, and up until Doom, had never published anything
themselves.  id has published through a few companies, Apogee being one of
them.  As mentioned before, Keen Vorticons, Keen Galaxy, and Wolfenstein 3D
were distributed by Apogee.  (At later dates, id went on and also published
Keen Galaxy and Wolf3D in the stores through GT Software, now called
GT Interactive, but that's another story.)  Some earlier id efforts such as
Catacomb 3D are distributed through Softdisk.  (Technically, these early
games, such as Catacomb 3D, Dangerous Dave, and Rescue Rover, are Softdisk
games, not id Software games, though they were made by the same people who
would ultimately found id Software.  The first game released as an
"id Software" game was Keen: Vorticons.)

Starting with Doom, id has been publishing on their own. Apogee has absolutely
nothing to do with Doom, nor future titles by id Software.  Furthermore,
starting with the release of Heretic, id Software started publishing games
on their own label.  They licensed their Doom engine to two companies, Raven
Software and Cygnus Studios (whose credits include Raptor).  (The game
produced by Cygnus Studios was cancelled and picked up by a company called
Rogue.) The two games produced under this agreement (Heretic and Strife
respectively), were published under the "id" label exactly the same way that
id Software themselves published Wolfenstein 3D under the "Apogee" label.

Twice, Apogee has licensed id's game engines for its own use.  In 1993,
Apogee licensed the engine used for Commander Keen: Goodbye Galaxy! for Bio
Menace.  In 1998, the Quake II engine was licensed for use in Duke Nukem
Forever, although this was later abandoned in favor of Epic's Unreal engine.
}Still later, the production of Prey, which had been in development at
}Apogee/3D Realms for years, was transferred to Human Head Studios, which is
}using the Doom 3 engine.

There was no "big divorce," or "feud" or anything along those lines
between id and Apogee.  The id crew and the Apogee crew are all good friends.