At the moment I own both. In some ways they are similar, but I could make a case for both! I love the x100 as it takes really good quality pictures and it's always in my pocket. 60% of all my "exhibition" images taken in the last 18 months have been taken with the x100. You don't have to decide which lens(es) to carry, there is just one, and it's a great lens. Its silent in operation, so very discrete.
I've just bought the XE-1 - it's faster, the resolution is better, and the viewfinder is better, but that is good, as it's the only option. The X-pro has an optical finder, but that tiny bit of extra bulk means it won't fit in my pocket. I have the 18mm lens, though at the moment it's got my Zeiss 50mm f1.4 on it. I think I will buy other Fuji lenses for it as they are amazingly good for the money.
I am selling my x100 as the XE-1 does everything the X100 does, and I'm probably selling my Canon 5d as I think the XE-1 will do everything that does for me too.
A lot of what makes these cameras so good is difficult to put into words, you really need to handle them - they just feel like they have been designed and developed by photographic enthusiasts. They remind me of my much loved Contax SLRs - solidly built tools that just do the job.
I know this is blasphemy, but having fantasised about a Leica M9 for some time, I finally had a chance to use one. I had the best part of £8000 of kit around my neck, and it felt heavy, cumbersome and rather too much like hard work to use. The x100 felt much more "part of me". Saved myself a small fortune I hadn't got anyway!