There are lots of differences in organs. As above, stadium organs are derived from the theatre organ. But it goes much wider than this. Even in church organs you have minimally German, French, and English voicing. Then some instruments are built for concert use and not for devotional use, but will still often follow a given school.
An organ designed to provide background music in services and to accompany a choir needs a different set of core capabilities to an organ designed to play front and centre in a concert setting, and even religious differences can influence the music and the desired sound palate.
Pipe organs have a pipe for every note at every different stop. The design of the pipe gives the stop its sound.To a very large degree no two pipe organs are the same. Budget, size constraints, use case and taste dictate what is built. From the huge range of possibilities an organ can be crafted to meet your desire.
Theatre organs add a range of “toys” or sound effects and other instruments allowing the organist a wide range of adornments when accompanying a movie. Drums, horns, xylophone, etc. Theatre organs try to provide a very wide palate of sounds. In their heyday they were trying to provide the organist with the ability to play almost anything appropriate. So they do include the capability of creating a church organ like sound. Not to the full extent of a dedicated church organ, but well enough to convey the idea. Theatre organs can become insane in size. As can stadium organs. The biggest organs being the Boardwalk Hall organ in Atlantic City and the Wanamaker Store organ (which is the largest in the world depends upon how you define it.) Another feature of theatre organ sound is tremolo. They can use a dedicated blower to vary the pressure across a wide range of intensities and speeds. Traditional organs may incorporate a temulant, but its effect is very muted in comparison with the capability of the theatre organ. That heavy tremolo is a big part of the theatre organ sound.
Organs can have different internal operation as well. Theatre organs came along after electrical systems were generally available for operation, so many are electrical or electro-pneumatic in operation. With switches under the keys and banks of switching logic attached to the stops to control the speaking of each pipe. Traditional organs are mechanical, with tracker rods, levers and rollers directly actuating valves in wind chests, with the stops selected by mechanically actuated sliders (little more than slabs of wood with holes in them) on the wind chests. What is interesting is that mechanical organs are still in favour as they provide a level of connectedness between the feel of the key and the opening of the valves that is missing with electrical systems.
Another thing about theatre organs is that they came at time when wind for the pipes was supplied by electric power. Whereas not all that long ago pipe organs were pumped by humans. Sometimes many humans.
The new technology meant that much higher pressures were possible, and organs could add much louder and more powerful sounds, especially horns. The Wanamaker Organ boasting the record loudest musical instrument known. Church organs were limited by the ability of the organ bellows to deliver wind both in pressure and volume. The available sound palate was thus also limited.
Traditional organs followed the music (and the money) and have traditionally been for devotional music in churches. The musical use case is different, and the sound palate different.
The basic sound of a theatre organ is different. Not inherently, but as a matter of taste and musical use case. The popular music of the time must also have been a big influence. Theatre organs came about when radio had spread music to the masses, and a revolution in popular musical styles was upon us. So they choose a different set of sounds from the available palate than devotional or concert organs and invent many new sounds.
