The reason I went with the core bit (certainly not expense) is that I wanted to be totally unplugged. With 3 charged batteries, I can drill between 25 and 30 holes, which is plenty for the sessions I do. I do need to run down to the quarry after every two or three holes to refill my water, which slows me down a little. I don't mind the break in the drilling--besides, I can have a cookie and think of my good fortunes to be in such a beautiful place.
The core bit, I believe, is used a lot for drilling holes in countertops and leaves a very finished surface--not important to me--whereas the masonry bit, I would guess, would leave a more rugged surface. I've attached a picture of a piece of a core.
I've also read that it's real important to wear some respiratory protection--especially with the masonry bit as the dust can be quite harmful to breathe. The core bit just runs all the ground up rock out as slurry in the water coolant, but I wear a mask anyway...
I agree about not worrying about the holes. They will be a constant reminder of my hard work!!