After being significantly underwhelmed by both, I'm not sure which I'd prefer.
I prefer to remember the handful of highlights and try and forget the unpleasant parts. However, Oblivion probably wins because although it had less high points than FO3, it also had less low points (Little Lamplight in particular was a huge failure for me, in it's entirety, with the "you can't use that now" lever the cherry on top).

I'd say that Fallout 3 looks a lot better there. In fact Oblivion kinda looks like Morrowind with a high res texture mod there.

I played both extensively on the PC - both vanilla and modded; and enjoyed my time playing both.
I prefer Fallout 3 (much to my surprise )

I enjoyed Fallout 3 a lot more.

I'm neutral on both. I enjoyed Oblivion since I was a big fan of Morrowind, and Fallout 3 is epic on it's own level.
Oblivion had more customization and brighter graphics (Which I loathe), decent combat, and varied locations. The one problem is that some monsters are ridiculously strong at some levels and it gets repetitive.
Fallout 3 is darker and more violent IMO. Combat is excellent, story is awesome, and locations are epic.
Your choice.

Fallout 3 has guns.

Fallout 3 had its moments. Oblivion had none. Honestly, I couldn't make it past an hour of the game. I've invested around 20 into Fallout 3. Its nothing like its predecessors, but its not horrible.
So I give my vote to Fallout 3.

Well I only picked up Oblivion GOTY yesterday (au$25 from EB, bargain!) and from what I've played so far, there's not a LOT of difference in the feel of both. In all fairness I can't comment on the story or quests since I've not really played Oblivion enough to get a feel for that side yet
Fallout 3 feels quirky and interesting where oblivion feels fairly generic and a bit dull
Oblivion has half decent melee with effective blocking where Fallout 3 involves running backwards and shooting (seriously, until I got my critical hit ability to full I'd have covered as much ground backwards as forwards)
Fallout 3 has reasonably good menus and controls whereas Oblivion feels clunky and more of a direct PC port (sounds like a weird thing to complain about but the menus feel deep rather than wide and kind of feel like you need a mouse when trying to pick stuff up because of small hotpoints)
Oblivion has colour where Fallout 3 is brown (understandably so what with the nuke and all)
Fallout 3 has Liam Neeson where Oblivion has Patrick Stewart (you have to decide for yourself which one wins there)
If you don't have a burning need for one setting or the other (have a look for the opening section of both games on youtube), I'd say grab whichever you can get the cheapest

Funny thing about the menus are PC gamers were complaining of how consolized that were with only a few items at a time showing up and such. I guess they just screwed up the menu UI on both ends trying to please everyone rather then make two different designs using the strengths of each system.

As a huge fan of Fallout, Fallout 2, and Morrowind, I have to say I was really excited about Oblivion and Fallout 3 (pre-ordered both). However, I ended up being much more enchanted with Fallout 3.
My big issue with Oblivion was the same-ness a couple people have mentioned. I liked the unique feel of ruins in Morrowind (i.e. one Dwemer ruin to the next held different items/different floorplans). In Oblivion, not only is a lot of the loot more meh, but most ruins feel and look the same.
Fallout 3 has less content, but is a more cohesive and, I think, memorable whole. Both are great games, though, and I recommend you work your way through both at some point. But I'd recommend Fallout 3 most of all.

I own both and all exspansions for both and over 20gigs of mods for both...
ive played (playtime) oblivion into the multiple hundreds of days mark....
ive played (playtime) fallout for only a few months....
O- has more going on for me, alot of story, alot of fun.. and the mods are massive and make the game live long into its end times. ive modded the hell out of it, created deep sprawling dungeons and just relaxed in the game. Ive played over 13 classes and had fun with them all playing out a full roleplay experience.
F3- there is no uniqueness, it feels so much of the same... it doesn't really pull me in and there are a lot of crashing issues. and there are no where near as many things to do as Oblivion. As for classes and differentiating them.... Mods are the only way you can stand out, EG: Roboco Certified allows you to hack robots and make then your pets... there is another that allows you to call troopers... anyways its really had to do anything but small arms and survive. everything is to scarce or hard to come by. in the end your always in the same type of gear.
If you want a Post Apoc shooter, get S.T.A.L.K.E.R. Call of Pripyat... Shadows of Chernobyl was amazing as well but... CoP is a living cast of all things that are amazing in all time. Ive yet to play Boarderlands but i hear good things... until a repack of all the extra crap comes out and hopefully without the 0-day DRM they snuck in i wont touch it.
I guess if i had to rate the games 0-100
Oblivion = 98
Fallout 3 = 70
Big Rigs: Over the Road Racing = 5

Yup, running backwards was quite annoing, but i spend about 90 hours in game just to find as many things i could. after 20 h there is not much to do than walk around and shoot everything that moves.

I find FO3 much more enjoyable than Oblivion without having to download a few gigs of mods to add some spice to locations and enemies and make it interesting enough for me to play

Oblivion sucks less.