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Many families of rockets use booster rockets, most notably the R-7 rockets that the Soviet and later Russians use. In comparison the American space-race era rockets mostly didn't.
For example, the Saturn V was an enormous rocket that might have been easier to make smaller with boosters instead.
What were the reasons they went with this approach?
There isn't that much of a difference after all.
It's mostly a matter of perception because the R-7 family evolved into many different variants, so it feels like most Soviet launch systems use liquid boosters. Proton, for example, doesn't use any strap-on boosters.
The following "early ICBM" designs of the early 60's are actually pretty similar:
Essentially the only difference here is that Atlas doesn't drop first-stage tankage, and that's because the light balloon tanks made that unnecessary.
Both designs were motivated mainly by concerns about reliably air-starting a bunch of big kerolox engines. It's way simpler to ignite everything on pad and then just drop stuff off, even though it's strictly less efficient as part of the second-stage tankage is already empty by the time the first stage burns out.
However, in the end, the R-7 family ended up spawning most of the Soviet space program, while the Atlas was just one of many competing designs among US launch systems.
