I have several XT1 bodies, and my colleague shoots with a bunch of X-E2 bodies. In my experience they are very similar in terms of capabilities and features. The major difference is the body style: the X-E2 is a smaller "rangefinder" style, and the X-T1 feels like a small DSLR. The X-T1 electronic viewfinder is a tad better - larger, anyway. Autofocus is almost exactly the same - good single AF, not so good on moving subjects. Image quality should be pretty much exactly the same. The X-T1 has more external controls using dials and buttons, especially for aperture and drive mode.
Honestly, if it were me and I were dipping my toe in the water with the Fuji system, I'd get the X-E2 and put the rest of the money towards a couple of good prime lenses. All the Fuji lenses are good, but in my mind the smaller primes are the reason to shoot with it. The 14/2.8, the 35/2 or the 35/1.4, the new 23/2, etc. I happen to love the 18/2, though it's one of the less beloved lenses in the system
Hope you enjoy it.