My Woodsman is extremely reliable with a broad spectrum of ammo, at least as reliable as my Buckmark, and probably slightly better. Trigger is also slightly better. Mechanical accuracy is a wash -- both are very good. Fit/finish is no comparison. Pull the slide on a Woodsman and you know some work went into that fit.
BTW, the second series Woodsman had a push-button mag release, if that is important to you. The first and third series have the heel latch.
Personally, I have a strong preference for the first series Woodsman -- the original John Browning design. The reason is the size, weight, and basic economy of the design. The second and third series were somewhat bulked-up in the grip area, making them more comparable to modern pistols.
This illustrates my basic problem with the Buckmark:
It's a fine-shooting pistol, but there is nothing size-efficient about the design. It's a .22, but you might as well be carrying a .45. As a range/target gun that issue won't matter, but it becomes quite obvious when you start carrying it around outdoors.
The first-series Woodsman, by contrast, will slip into a coat pocket. The grip frame is no larger than it needs to be. The gun just feels like it is scaled perfectly to the .22 cartridge.
For me, the one fairly serious drawback is that it's a work of art from the 1930's, and I experience a guilt-factor associated with high-volume blasting through the thing. Shooting and outdoor activities are not quite as care-free as they could be. For that reason, when I want to carry and shoot a very reliable, size-efficient, and highly shootable .22 outdoors these days, it's usually a Beretta 71.