ramdog1;
             Either rifle would be legal to shoot in an ASSRA match as long as you used cast bullets with no gas checks.
             The rule book states "single shot rifles-non bolt with lead or lead alloy bullets [no gas checks ].
              Welcome aboard ramdog1------leadball

Thank you to Leadball & DonH for the info. I would have to say that the Browning is the most accurate rifle I've ever shot, I love the gun. But I've killed plenty of deer w/ the Remington too. Nothing like a single shot, one chance to get it right.

Ramdog,
To answer your question about the 7x57 in a Roller with modern ammo. This concern came up when these older Rollers showed signs of excessive headspace, and even some heads splitting from the case in worse cases.
The design of the RB in a modern cartridge like a rimless centerfire like the 7x57, can create some ugly situations if the headspacing is a bit too much. More so than other designs such as a falling block, bolt action, or any other strong type action.
If the action is in good shape it wont be a major concern, but still there is the possiblity that it will eventually develope excess headspace in such a high pressure cartridge. 
I had one of the old 7mm Rollers when I was young, and never fired anything but factory ammo in it. Never knew any better until it split a case one day, and gave me a peppering in the face. That's when I started reading and asking. I felt sorta stupid, as everyone I asked seemed to think everyone else already knew this info. Well duh!!! Not me!