Tough call. I'm sure my answer would change from day to day. These were two of my favorite EDC knives even before I got the custom scales on my Mini-Grip.
It's pretty much a draw on all categories. Both 154CM and VG10 are excellent steels, and Benchmade's and Spyderco's treatments of each is top notch. The 556 has a high flat grind, and the Delica has a FFG option. Both are excellent slicers. Company-wise, both Spyderco and Benchmade are reputable and have great customer service.
Comparing the knives themselves, the ergos on the Delica are more comfortable. That's not to say the Mini-Grip isn't comfortable, but the longer handle combined with the contours of it and the blade ramp does provide a more comfortable grip. I can get a decent 3 to 3.5 finger grip on the Mini-Grip, versus a comfortable 4-finger grip on the Delica.
Lock-wise, the Axis lock has the advantage. Quick to open and close, and a solid lock-up with no movement at all. The back lock on the Delica is strong and secure, but if you are cutting with a fair amount of downward force, there will be some movement. That appears to be a limitation of the back lock. It doesn't affect cutting ability or the security of the lock-up, but under some situations, you will be able to feel the blade and lock bar move. Both can be opened and closed one-handed, although the technique for the back lock can take a little more practice.
As for pocket carry, the Mini-Grip does have a smaller footprint in pocket. It's slightly wider at the thickest point (since the handles are a bit rounded), but it is shorter and narrower. Neither has deep pocket carry (although deep carry clips can be obtained separately for either), and both expose about the same amount of the knife outside the pocket. The Delica does have the option for tip-down carry, versus the Mini-Grip is tip-up only.