I recently went through this for both myself and with a friend. I'm already in the Fuji camp with an X100 and am extremely pleased with it. Here's a few of my thoughts:
I expect both to be very good cameras. Well built and hopefully without quirks.
I give the lens nod to Fuji. For me, the line makes more sense. I shoot primes and a 35 and 50 equivalent is fine for me. The 35 will be out next year, until then I have the X100 if needed. Fuji's lenses are good across the line, the same can't be said about Sony's. There's the Zeiss 24/1.8 at $1,000+ when you can find one for sale and the new, untested, 35/1.8. I like the new kit zoom but I prefer the Fuji speed and optics in their kit zoom.
I have the advantage of shooting a NEX-7/CZ24 for 3 weeks this summer. I did not purchase it as a result of the evf had too little eye relief, the transition from rear display to evf was quite slow and the CZ is heavier than it needs to be. While I could live with the control set, I'd take Fuji's in a heart beat. Metering in the X100 is superb, somewhat better than my experience with the 7. Auto white balance in the 7, under artificial light, was poor. Metering to me is important.
AF speed at this point is an unknown. The pdaf approach Sony is taking is the same as Canon took with their mirrorless interchangeable. The Canon is not quick, I have yet to see a definitive test on the Sony. However, my expectations is it will be snappier than the Fuji with the current lenses. Once the zoom, the 14 and the 23 are out, who knows.
I prefered the NEX-7 in hand to my X100. I expect I would prefer the NEX-6 over the XE.
There's a lot of concern about post in the X Trans world. First, most of the issues are with Lightroom and the fact Aperture does not support X Trans. For LR users, render in SilkyPix and import to LR. For Mac users, render in RPP, which has an Aperture plug-in, and import. There are those who won't/can't leave their chosen app. I was never like that, when I shot Nikon I rendered in NIK, never LR. For me, the raw issue is a non-issue. And the jpegs are to die for.
Last is how I feel about the camera and the company. Fuji is a quirky company with a very small, but profitable camera business wrapped up in a profitable multi sector company. Sony is an unprofitable camera business wrapped up in an unprofitable multi sector company. Fuji knows film, $160,000 video lenses, Hasselblad and Phase 1 bodies and lenses and some specialized optics. Sony has a superb sensor business.
As far as the camera, to me the Fuji exudes charm and a bit of classic elegance. The Sony is a nice functional mass market camera.
Cost wise, with the CZ and the kit zoom, versus the 35/1.4 and the kit zoom, it's about the same. Though with different optics, the Sony can be a lot less coin.
I have an XE on order with the kit zoom and the 35/1.4. My friend is still on the fence.