To me it's all about level design. Old School FPS's might have had a theme and atmosphere, but they tended to be very gamey and arcade like in their level design. They had intricate level design for its own sake, including things like mazes, floating platforms, massive arenas, implausibly massive, labyrinthine complexes, and etc. They also tended to contain relatively few art assets, but tons of levels. Gameplay and "Wouldn't it be cool if....." always took precedence over realism in terms of level design. The city level from Doom 2 is a great example. It's a massive level which looks vaguely like a city, but it's more or less the same 50 art assets used over and over, just mocked up in the form of a city. Nothing functions or looks like a real city. It's all about gameplay and using the outline of a city to do fun level design type stuff.
In contrast, new school RPGs (probably beginning with Dark Forces 2, but finding the first instance of the formula that would ultimately dominate the genre of FPS for a decade and a half in Half-Life) allow realism to take precedence over level design. So while level design can occasionally be complex (especially in the early examples of "realistic" FPS level design, like "Dark Forces 2") there's nothing that will break the immersion of the experience (no endless colored keys, floating platforms, or etc.). Likewise, the level design tends to follow a narrative arch of sorts, with the character progressing through the levels, changing locales, in a linear fashion. This lead, in its most extreme form, to the "rails" shooters like "Call of Duty" which were basically just a bunch of linear set pieces.
Presently, though, I actual think linear rail shooters are outmoded (which is probably the main reason why Valve has no interest in doing Half Life 3). Now the FPS is dominated by open world "Far Cry" style games and RPG-lite type experiences.
There are also other elements, though. Old school tended to privelege gameplay over realism in all areas, including: number of enemies on screen at once, number of weapons to carry, speed, power ups, and etc.
Examples (Codex will no doubt bitch and disagree with individual examples, but that's the fun of lists):
Old School:
Heretic
Doom
Wolfenstein
Rise of the Triad
Painkiller
Duke Nukem 3D
Serious Sam
Hexen (although moving in the direction of a hybrid game)
Hybrid (transitional games, or games with elements of both):
Dark Forces
Dark Forces 2 (probably the best example of the best of both worlds: old school labyrinthine level design with new school realism)
Samurai Warrior (both original and remake)
Blood
Thief
Stryfe
Realms of the Haunting
Hexen 2
New School (linear):
Half-Life
Half-Life 2
Call of Duties
Deus Ex (all of them, although some of the original's levels were very large)
Bioshock 1, 2, Infinite
Dishonored
New School (open world)
Far Cries
Crysis
Stalker