I have neither, but have been doing research on the same subject and I wish there was a definite answer; kind of a Ford vs. Chevy question.
I'm looking at a Buckmark Camper, or a Ruger 22/45 5.5" Blue, bull-barrel (both low-cost, entry level .22 cal. pistols). The 22/45 has a different frame material (Zytel polymer) and grip angle than the regular Mark III, like a 1911 grip angle - possibly a plus to consider if you are going to shoot center fire pistols later on. From what I can gather, either one will perform well and reliably for you with most ammo. It is best to shoot each first, if possible, to see what fits your hand best; if not, at least go to a store and hold them to see which feels the most comfortable. Many people say that the Ruger is a pain to field strip and clean, but after watching a youtube video by jimmybuzzkill, I find that hard to believe. I couldn't find a good video for the Browning that was as well done, or made the stripping seem as easy; maybe I missed something there.
AFAIK: the Ruger comes with 2 mags. is drilled & tapped for a Weaver scope base adapter which is included. The Buckmark has one mag. and is not drilled. The Ruger seems to have an advantage in availability for after market upgrades if you want.
I still haven't decided as I hope to shoot both soon as I can arrange it. I'm sure others will submit more advice too.
Good luck.