For curiosity sake, and to satisfy some members here, I just took some shots w/ both lenses set to as close as possible settings at the longest matching focal length and aperture, which meant trying to set the 16-55 II set to 50mm (close I could get was 49.8), and f/4.8 (close I could get was f/4.5 or f/5, so I went with f/4.5).
Note that there may be a possibility that I have a de-centered lens, who knows, this is just one sample, and I'm not a professional tester, so don't base any decisions off my stuff here. I'm sure we'll eventually get a more precise review from someone more reputable

I compare/test my lenses shooting around for a few weeks and build up my observations in order to notice trends and the character to get a holistic picture, it isn't just one aspect that makes/breaks it for me.
Center of frame: lenses are very close, and the 16-55 II does better here compared to its 55mm, but the 16-50 is still noticeably sharper.

16-55 II vs. 16-50, center
Edge of frame: 16-50 is much sharper, no contest.

16-55 II vs. 16-50, right edge
Ok, so the "kit" lens is sharper than the newest Red-Badge lens, but here's where I think the trade-off happens: bokeh quality, and probably a smoother rendering. The 16-55 II's bokeh quality is substantially better (in my opinion, of course, since this is all subjective), and so perhaps the lens designers may have sacrificed some ultimate sharpness in exchange for more pleasing performance in this area.

16-55 II vs. 16-50, bokeh near center
Open the 16-55 II to f/2.8, and it's clear what you're getting with the constant f/2.8 (or sacrificing when opting for the 16-50):

16-55 II @ f/2.8 vs. 16-50 @ f/4.8, bokeh near center
My personal conclusion so far after shooting with the 16-55 II for a month:
This 16-55 II is my first experience with a Fuji Red Badge lens, and I honestly expected it to be a world beating, test chart eating engineering masterpiece, but so far the performance I'm seeing isn't that. HOWEVER, sharpness, MTF charts, brick wall tests, etc, aren't everything, and I think maybe the design philosophy for this lens was meant to encompass more? I love lenses with character and am OK with imperfections, my prime lineup is the Fuji original trio of 18 /f2, 35 f/1.4, 60 f/2.4 squarely for that reason, so I'm definitely not after absolute sharpness and perfection, just surprised at the results is all.
It's also a good thing that Fuji isn't pulling punches with its kit lens just to protect the big boy Red Badger, it's letting the spec features/limitations define the market... if you need constant 2.8, extra 5mm, and click-less aperture, you go 16-55 II, but if you don't necessarily need those features then the 16-50 is an easy choice, and you can be confidant knowing you're not giving up anything performance-wise.
So far, I'm leaning towards keeping the 16-55 II and parting w/ the 16-50, as I'm wanting a 1-lens solution, and have more need of a lens that can work indoors vs. one that's better suited for outdoor/landscapes.