The original Sharps CO. closed it's doors in 1880. Shiloh started out as the Shiloh Rifle Manufacturing Company and was owned by Wolfgang Droege and goes back to early 1970s late 60s? C. sharps was the main distributer and moved to Montana in 1981 and Shiloh followed in 1983. In 1991 Wolfgang retired and the Bryan Family took over. Shortly after Shilohs move the two went there separate ways and the word is things are a bitter between them. This is pulled from Mike Venturino's book "Shooting Buffalo rifles".

"Accuracy-wise, does one have the edge on the other?"

I dont think one realy has a significant edge on the other both rifles are capable of dropping 500+ grain bullets into the 20" 10 ring at 1000 yards open sited.

After doing the research it seems that barrels using cut rifling are considered to be of higher quality than the button type. And like anything I'm sure this can be argued all day long. Here is an artical on Barrel making
Cast vs milled action again that can be argued all day long both certainly perform well. Rifles like anything else are only as good as the sum of their parts. Personaly I think C.Sharps uses slightly higher end components. In the end it boils down to personal preference.

Regardless which one you buy you won't be dissapointed.