They are quite different. The RH is an older design, employing partially lugged barrel and a single coil spring to both power the hammer and return the trigger, a la the GP-100's predecessors. My 5.5" .45 Colt SS RH, with it's original spring, was still tempermental as to primers, requiring my Federally primed loads - and still giving occasional ftf's. The trigger pull seems to be a bit long, too... and no speedloaders will work with the stock grips, as they haven't been dished out. The Ruger scope rings mount to a rib running the length of the barrel on 'scope ready' models. They were available in 5.5" & 7.5" lengths in various calibers, with only the .44 Magnum now being available.
The .44M SRH, like the .454 & .480 versions, is an acquired taste. They have a simple, but very rugged, barrel protruding from the frame, making for a bit less traditional 'look'. The scope rings mount on the long frame top. They have a better trigger, utilizing the same fine lockwork as the GP-100's. They will allow HKS speedloaders to be used. I liked my 7.5" .454 SRH, which I had for years, but chose to keep my newer 5.5" .45 RH, as it grouped better with the wimpy rounds I shoot. If I wanted a Ruger .44M DA, I'd opt for the SRH... which is a couple ounces lighter, oddly, in the 7.5" variant than the RH. The backstrap enclosing absorptive stock rubber grips can be gripped higher - and offer better recoil control - than the RH's wood stocks.
Actually, when I needed a .44 Magnum, I opted for a S&W. Their 629's are fine for my uses - and what a fine trigger!
Stainz