I have worked the Nex-7 for about 18 months, the Fuji X-E1 for about 3 months.
I like the layout of the camera controls and the LCD, (both resolution and ability to tilt). Sony's EVF is excellent and their implementation of focus peaking is very well done. I mainly shot raw on the NEx-7 and found that mine was about a iso 100 to iso 400 camera, as much higher and the amount of noise becomes pretty harsh. In a push maybe iso 800. You can get an excellent read on the noise from the Nex-7 on the dpreview comparison pages. The actual implementation of menus for control of various features to me is confusing still.
AF with the emount glass very good, AF with the Sony A adapter (the LA-EA2) is also very good. I tried several of the dedicated e mount lenses, but only stayed with the 18-200. My main carry lens on the NEx-7 was the Sony 16-80mm with the LA-EA2. Not a small package, but one I found I liked.
I never found a wide that held good resolution to the far corners of the sensor, including the Sony 10-18mm. The Sony 10-18 showed strong magenta cast in the corners on my camera @ 10mm (15mm) and since I wanted it mainly for the wide side I returned it. No wide lens I used on the Nex-7 ever seemed to hold to the corners for details and sharpness, the Zeiss/Sony 16-80 was the best I could find and even it had trouble at 16mm (24mm equivalent). As the sensor is 24mm, I often found myself cropping it down to the same as a 20mp sensor due to corner softness/detail smearing in the corners on wides. I never tried the newer Zeiss 12mm.
Excellent video platform, very easy to use, I never upgraded the firmware to allow the video button to be taken off line and it does get in the way many times.
I only used LR for raw conversion and liked the files, but many times over the noise above iso 400 caused problems.
The coming 24mm FF may be the answer as this chip has always been a cleaner all round chip, however noise tests I have seen from the RX1 did not impress me.
The Fuji has been a welcome addition. I am still constantly impressed with just what the X-E1 can deliver.
Lenses, I have the kit 18-55, and the 55-200 and have briefly used the 14mm and 60mm macro. All of these lenses showed excellent resolution, especially the kit 18-55. This lens at 18mm holds very good all the way to the corners. The 14mm also was impressive to the corners.
AF, many complain about it. With latest firmware, I find the AF OK, it does have problems in low light and is not the best for action or continual low light shooting. Manual AF is easy now since peaking as been added. The X-E1's LCD is just OK, but the EVP excellent. If you don't mind using the EVP for shooting, the Fuji is a great solution.
Noise, much better, easily the cleanest camera in this class I have shot. Noise in the iso 200 to 800 is pretty non existent and you can take the camera to around 3200 before things really get out of hand. The lack of iso 100 when shooting is raw is a problem for me at times. Fuji's enhanced DR shooting does seems to help also even though you are at iso 400 and 800.
Raw conversion, I was concerned about this after reading so many bad posts. Actually I was pleasantly surprised as LR 5 and Capture One both seem to work fine on the files. The only area where I sometimes can run into problems are finer green details, mainly pine needles. I don't see the over processed look so many people complain about. Fuji's own raw conversion software IMO is a total waste of time and it might have benefited to work closer with Adobe and Phase One to allow them to tweak their software for this camera.
I miss the LCD of the Sony, as IMO it was best of breed and so far have kept my Nex 7 only for video work. I realize I have scaled back in overall output resolution but 16mp can often get the job done as well as 20mp. The Fuji is easy to setup for a stitching setup also.
I have not tried video on the Fuji, but it doesn't seem to be one of it's strong points from reading other reviews.
Night photography, the Fuji wins hands down. I never found a good intervoltmeter for the Sony, and it doesn't handle night shooting very well due to noise issues. The Fuji works well right along with my Nikon and Canon solutions and there are several intervoltmeters that work with the Fuji so you can setup for stacking.
The best solution is try them both if you can, as both cameras offer some excellent feature/function solutions.