The APD version lets in less light, so it is an f1.7 wide open for exposure purposes. This effect becomes less as you stop down.
The APD doesn't focus as well on the newer Fujis (although it makes less difference on the XPro1, which does not support the more advanced AF options). In good light it is fine and it is still usable in low light.
The APD costs a lot more.
So why would you want one? Because the out of focus parts of your image (the bokeh or area outside of the depth of field) are rendered better and the transition in and out of this area are better and hence the natural, organic look of the images and their 3D quality is better. The lens renders more like a good Zeiss lens.
Is it worth the trade offs to get better image quality? Only you can decide. For me, the answer is yes.
There are enough images made with each version that you can look at on Flickr. Once you look at 1000 from each lens, you will know if it makes a difference to you that matters. Look especially at images with areas of brightness in the background that is out of focus. If it is not obvious to you, I would say it isn't worth it for the three reasons above--price, AF and exposure.
To my eye, the Fuji 56mm R lens is not as good as the Nikon 85mm/1.4 or the Canon 85mm/1.2 on full frame. The Fuji 56mm/1.2 APD is better than the Nikon 85mm/1.4 and the Canon 85mm/1.2 on full frame. None of these is as good as the Zeiss 85mm/1.4 Planar, but that is an exceptional portrait lens, even for Zeiss. Don't get me wrong, the Canon and Nikon are both exceptional lenses. The differences we are talking about are relatively subtle. I don't think I would have purchased the non-APD version but the APD version is great.