Were they mostly just catching up with the US, or did they lead to developments we might not have made otherwise (at least as soon)? And I know they're different programs, this is really two questions.
The RD-170 engines that were developed for Energia are arguably the most advanced kerolox (kerosene and liquid oxygen fueled) rockets engines ever created. They generated approximately the same thrust as the F-1s that were used on Apollo but had a 20% better specific impulse (fuel efficiency, to over-simplify what SI is).
The RD-170s were only used a couple of times, but a half-sized version called the RD-180 has been used for 15+ years on the American Atlas V. ULA buys those engines because the US has no kerolox engines that can match their thrust and efficiency.