Ok, first we have to consider where you are going and where you have been. I've recently gone from 7D with the EFS 15-85, which is very similar in terms of sharpness to the 17-55. On a FF, I am seeing more detail with the 24-105. I haven't looked at the corners in painful detail yet, but in general, when I look at 1:1 ratio on the screen, there is a modest, but noticable, improvement. Please note, I still consider the 7D with the 15-85 to be excellent. Both are extremely capable. But I would expect you to see an improvement with the 24-105 on the 5DIII compared to the 17-55 on the 7D as well. So, coming from where you were, yes, I would expect an improvement.

However, as Busted notes, I've heard others comment that the corners are "soft" on the 24-105. Which to me just means that there is room for improvement on the 24-105 on a FF body. Unfortunately, if you are to follow TDP recommendations, there is only 1 zoom lens, the 24-70 f/2.8 II that is "overall" better than the 24-105. I've seen enough reviews to also think that the 24-70 f/4 IS is likely a little sharper, especially at the extremes (softer in the middle focal ranges), but it is also significantly more expensive.

....the next option is primes. So, you can enter the classic primes vs zooms debate.

Myself, I went with the 24-105 and am happy. Honestly, there are times when I miss the 15-85 mm. IS on the 15-85 is noticably better (4 stop IS on the 15-85 vs 3 stop on the 24-105) and I do notice the equivalent 31 mm lost on the long end. But the 24-105 on a 5DIII is still a great combo. But the best zoom, based on everything I've read, Bryan's recommendations, etc, for a Canon FF camera is the 24-70 f/2.8 II, if you can afford it.