You can't evaluate a purchase like this based upon image quality alone. There's much more too it than that and there just isn't a distinct advantage one way or the other, between these particular lenses.
For use on 60D, I'd get the EF-S 15-85mm. It just seems a better match for the crop camera. It's a reasonably compact lens on a fairly compact camera. The 24-105 isn't particularly big, but it is bigger and heavier than the 15-85.
IQ should be fine.... check out some of the reviews of the lens. There are a number of them on the Internet. The Digital Picture's review here
, allows you to do a number of head-to-head comparisons, such as their ISO 12233 crops and other image quality factors. You can compare the 15-85 directly with the 24-105 at 24, 35, 50 and 70mm... in some examples both on 60D.
To me it looks like the 15-85 has less chromatic aberration at 24mm, while the 24-105 has less at 70mm. But neither has a whole lot of CA. The 15-85 seems to handle flare better at the focal lengths they share. The 24-105 is known to have strong corner vignetting at the wide end (nearly 3 stops wide open at 24mm, one of the worst Canon lenses for this, though it's largely correctible in post processing or with Peripheral Illumination Correction in camera). But most of that vignetting is cropped away when you use the FF lens on a crop camera. The 15-85 appears to have slightly more barrel distortion at 24mm, but the two are nearly identical at other focal lengths they share.
The 15-85mm plus it's matched lens hood (sold separately) cost $670 on Amazon. The 24-105mm costs $1050. So you have to ask yourself, would $380 be useful toward something else you need? Or, is the red stripe worth an extra $380?
It also depends upon what other lenses you have. The 24-105 would fit into my kit better because I use a 12-24mm ultrawide and I use both crop and full frame cameras. But if you don't have an ultrawide zoom at all or have an 11-16 or 8-16mm, the 15-85 might complement the rest of your kit better.
If you plan to switch to full frame camera soon, or add one to your kit, the 24-105 might make more sense. But, frankly, I think full frame is vastly over-hyped here and on other blogs... Crop sensor cameras today come awfully close to FF image quality for most peoples uses.