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Image quality wise, the Fuji produces cleaner files with deeper color depth and probably slightly better dynamic range. I think the files from the OM-D have a little more detail in them. The Fuji has about a 1-stop ISO advantage and cleaner base ISO.
I did a quick ISO test here: http://admiringlight.com/blog/mirrorless-battle-om-d-vs-gh3-vs-x-e1/
As far as handling goes, the X-E1 is easier to pick up and shoot with due to the simple and effective controls (aperture ring, shutter speed dial, EC dial, etc).
The OM-D stomps the X-E1 in overall responsiveness. The AF is significantly faster, especially in low light, and the OM-D will lock instantly on targets that the X-E1 will simply fail at in dim light. Shutter lag is lower on the OM-D, and the viewfinder is better in very bright light as well as very dim light. In overcast situations or bright indoors with a good lens, the X-E1 finder is a little better. Overall, I prefer the OM-D finder.
The X-E1 has overall better image quality I'd say. As a full camera and system, the OM-D is the stronger candidate. There are some big holes in the X-E1 operation as it fits with a modern camera system, and there are only really minor things with the OM-D.
I own both and use both regularly, and if I could only keep one, it would be the OM-D without a second thought.
My full reviews of both: