I have a fairly un-biased opinion. I have worked a fair amount on both the FJ and the JK's (newest was 2009).
Interior
Both have uncomfortable and ugly interiors, but thats the "cup holder prevails" world that we live in. the JKU has more usable storage space. I do like the rubber mats in the FJ but for the price I think carpet should be an option.
On Road
Driving the FJ is not bad it has a nice amount of power and the transmission is responsive (keep in mind I drive primarily mountain roads), steering is great as it should be from a rack and pinion. Visibility is like driving a chop top cargo van, I always find my self leaning forward because I feel like I am missing somthing, Also the blind spots (everywhere but out the drivers window) are killers. My mom owns an FJ and my Dad has backed it into two parked cars while parrallell parking (I vouch that its not that he is just a bad driver, hasnt has any issues like this in other vehicles).
The JK drives like a pickup and in my opinion it should. The motor is gutless (haven't driven the 3.6, I hear it's much better). The transmission (auto) sucks, always hunting for where it should be (again mountain roads). steering is responsive but not as "tight" feeling as I feel it should be (similar feelings with my dodge 2500 CTD).
Suspension
The suspension design of the JKU for off road is superior, but its not without its flaws. I have done repairs on three with under 70k miles on them still bone stock that had major death wobble issues. The ball joints and pannhard are common problem areas, even in stock form (especially Rubicons).
The FJ is smooth on the road and is nice on those long fire road that are just annoyingly bumpy. there isn't much in the articulation department but its well designed and doesn't have the maintenance issues that the jeep does, although with heavy off road use you will be replacing TREs and Ball joints more frequently.
Drive train
for the sake of this being shorter I will stick with the Rubicon model. the motor is gutless and has a few reliability issues ( I know of a few that have spun bearings while still being well under 100k miles)
transmission, manual is good, auto is typical Chrysler not bad but not great defiantly could benefit from an additional cooler (again mountain roads and some light towing)
transfer case, good reliable as long as you keep the fluid level up. most common failure I see is: the cust says "i thought it got its fluid from the transmission" I say "no" He says "that sucks" and pays to have it rebuilt. (this was a non Rubicon so not a 241or)
drive shafts are, IMO, Weak.
Front axle can be bent even in stock form from abuse but is generally fine
rear axle is all gravy
electric lockers are nice but I have had minor issues with them.
FJ motor is awesome, typical Toyota reliability. nice power
trans is good only had one in our shop to rebuild and it had 170k on it and went through a deep water crossing, and apparently it was thirsty because it took a big drink.
transfer case would be nice to have an option for a 4 to 1 but in stock form it does ok.
front axle is typical for IFS setups prone to failure in the right (or wrong) conditions leaves me wanting more
rear axle is tried and true and the TRD locker works excellent
Reliability goes to the FJ hands down
Aftermarket support is there for the FJ but no where near as prevalent as for the Jeep.
For those of you who say "I prefer to buy American" keep in mind Jeep is Fiat (defiantly not American). Most of the Toyota we buy here in the us are built here in the US (not sure of the FJ) which creates a lot of jobs for our countrymen (and women). Not sure but I think jeeps are assembled in Mexico or Canada (as the dodge trucks are). Think about the fact that Toyota has been true to its design and has produced a quality product for the US since the early 70's (could be late 60's don't know my Toyota history that far back) The jeep name has bounced around many different parent companies which hasn't always benefited the end user.
all in all if I were shopping for a new overlandingish vehicle that I could DD, and I had to choose between these two, I would buy the Jeep mainly because the FJ is hideous (I guess I'm just a little vein). I think both can be built into respectable offroad capable vehicles, The JKU will be more capable for less money though.
In all reality I was asking my self the same question a couple months ago between a Tacoma 4door or a JKU and the wife and I decided on the taco (keep in mind I live 30min from the Rubicon trail) for the sake of reliability and a lower cost of maintaining it. Of course life happened and we didn't get either (I picked up a wrecked TJ instead and I'm rebuilding it).
My only other suggestion is make a very informed decision, go test drive as many different vehicles as you can even in the same make and model, even if its the same car one will feel different. Go with your gut and don't let sales men BS you into thinking one is better than another. Always go to the dealer around 10:30 am so you have an excuse to leave for "lunch" and think over the arrangement without them circling like vultures.
I hope my opinions are helpful in some way.
See you on the trail
Nick