Winchester brought out 3 cartridges in the Model 1894 lever action rifle:
.25-35 WCF
.30-30 WCF
.32 Winchester Special
Remington, not wanting to lose out on the growing popularity of the new smokeless powder rounds, had John Browning design a semi-automatic rifle, the Model 8.
Remington also introduced four new cartridges:
.25 Remington
.30 Remington
.32 Remington
.35 Remington
The .35 Remington was the only new and unique cartridge in the line up; the others were simply rimless versions of Winchester's rounds. They were so close, in fact, that dies and loading data were interchangeable.
The .25, .30, and .32 Remington rounds soldiered on for some years, being chambered in Remington pump and bolt guns, but never achieved the kind of wide-spread use and acceptance that the Winchester rounds did.
The exception was the .35 Remington, which gained quick and lasting popularity, in large part likely because Winchester didn't have a comparable chambering for the 1894 (although a common gunsmithing conversion as the .35/.30-30, which was used to salvage guns with worn bores).