I've not yet figured out the difference between Doom series and Quake series, since the background storyline (or what is like that) seems to have no clear difference. When Quake was released, I thought the difference was the main character. Doom and Doom2 both had Sarge as the main character. Quake's main character was Ranger. Since Quake2 was banned in China for its publisher running into some trouble, I don't know who was its main character. And when Q3A was released, it was a complete multiplayer with Sarge as the default bot.

One big difference is that Doom uses peer to peer - so it was basically SP with MP attached. Quake uses dedicated servers, which you could say was MP with SP attached.

For peer to peer everyone has to connect at the sametime, no one can leave and (re)join mid-game.

For dedicated servers, people can leave and join when they wish and the server can take up a large part of the MP game network handling - basically it's what you see in most FPS's now.

Unfortunately, it seems the Doom 3 engine may again be peer to peer and so will Quake 4 (using the Doom 3 engine).  

For the story line, Doom follows the rift in ####, where as Quake deals with the invasion of the Strogg.

strafe jump ^^

Let's take this from the beginning.
In the beginning there was darkness...
And God said:
"Let there be Doom"... and there was Doom and God saw that it was great.

And it was great.

Seriously, Doom was THE game to revolutionize this minor genre known as the FPS genre and kickstart it. Quake is only a minor game in comparison, for without Doom there would be no Quake. Heck, without Doom, the FPS genre wouldn't be as widespread and popular as it is now.

As far as storyline goes, Doom has a much more detailed story than Quake: Scientists mess around with teleportation tech, it opens a direct door into the Underworld (I hate how you can't use the 'H' word - even though I know it's meant for the good of these forums) of devils (the place where bad folks go when they die according to Christianity, and the most totally evil place in the multiverse).
Doom takes place on Mars' two moons - the first one is still located in our dimension, but the other moon is sucked into the 'H dimension'.

Basically, the portals on both moons are screwed up, demons pour out of the gates and kill everything and Deimos (the other moon) is sucked into a wormhole that transports it into the Underworld. The folks on Mars send up their only troops to deal with the problem - you, the doomguy, are ordered to stay outside to keep your sidearm company (ok, maybe not) and the rest of the soldiers go inside and are wiped out, leaving it up to you alone to clear up the mess on the moon.

Btw, the main character in Doom is not Sarge - the Doomguy hasn't got a name (and if you suggest that awful name from the even more awful Doom novels I'm gonna punch you in the nose), but he's characteristic by that green combat suit and that cool space combat helmet with a closed visor.

In Quake, the storyline is embarrasingly similar (one out of many reason why I dislike Quake - it's too close to Doom while not capturing the true essence of Doom). But the overall difference is that Quake introduced true 3d graphics into the fps genre and had a better mp system than Doom. Another difference is that Quake has rather lousy single player compared to Doom, simply because Doom had original gameplay, while Quake pretty much copied Doom's gameplay (but losing out on a few neat details that Doom had, like corpses that were squished).

As for the storyline, Quake is about scientist tampering with teleportation tech (again), but this time it doesn't exactly opens a door to another dimension. In Quake the Evil badguy Quake is able to establish control over the teleporter slipgates and send his things into our world to steal and kidnap.
Humanity mounts a counter strike operation, but Quake attacks first - you're the only survivor and you're Ranger.

Plus, in Quake Ranger has to find and unite four runes, but that's just a minor element in the storyline.

And then there's Doom III, which is a much more detailed retelling of the original Doom game (a few story elements are changed to make it more realistic).

And then there's Quake 2 which has nothing to do with Quake except the name. In Quake 2, we have those cyborg aliens called Strogg that invade Earth and humanity launch a counter operation that takes the human soldiers to the Strogg homeworld.

And Quake 3 Arena has no story worth mentioning.