I own both and EDC the Mini Grip 556 mostly just because I subjectively like the looks better. I would frankly be self-conscious to use or lend a Delica in front of certain people that do not understand such an unusual, non-traditonal looking knife. But they are both excellent knives and neither is a bad choice.
Objectively, I would give the edge to the Delica because the grip area is longer and it is flatter in the pocket. The large Delica thumb ramp gives your thumb a more secure and powerful perch on the spine. You can also choose tip-up or tip-down clip placement with the Delica while you only get tip-up with the Grip. The Grip clip can be rough on pockets. Just a bit better ergonomics in general on the Delica. By the way, I would avoid the Delica with the stainless steel scales because they can get slippery.
One possibly significant difference is that you can easily wrist-flick the Grip open but it is virtually impossible to flick the Delica. Some people like to be able to flick, while some may prefer to avoid it because in some areas, that is a legal problem (NYC in particular, for instance). In general it is smarter to carry a knife that is more legal in more places and the Delica wins here.
Factors in favor of the Grip: I think that the Axis lock is more resistant to developing vertical blade play in the long run, but Delicas are so well made that I wouldn't consider this a deciding factor by any means. It is also easier/faster to close an Axis lock and possible to do so without putting your fingers in the blade path and I think that is just a good design principle (although again, not a deciding factor).
In the end, get whichever one speaks to you. Both work great.
If I were buying today, I would also consider the $40 WalMart Spyderco Native in S30V as a less expensive option in an arguably better steel (but no liners).