The Soviets undertook missions of exploration for much the same reason as the US did. For scientific reasons, of course, but also for reasons of national prestige on the geopolitical stage.
If you look at the Soviet manned spaceflight missions things are very much different though. Around half of the "Salyut" space stations were actually military "Almaz" stations (Salyut 2, 3, and 5), at least one of which was armed with a fully automatic cannon designed to engage other spacecraft. The Soviets also pioneered the development of anti-satellite systems starting in the 1960s, long before the US had even started an anti-satellite program. The kicker is "Polyus", the first payload of the Energia launcher, it was an 80 ton orbital warship. It was a prototype designed around the missions of anti-satellite/anti-spacecraft warfare and laying nuclear "mines" in space (in contravention of the Outer Space Treaty, of course). Polyus itself did not contain mines though it did contain a set of test targets and a 1 megawatt laser. But Polyus failed to achieve orbit due to a glitch during the launch.
As far as exploration, the US has contributed more to interplanetary exploration than any other nation or group of nations. It has sent more successful missions to other planets and been the only nation to send probes into interstellar space. Excluding spending on ground-based ballistic missile defense the total spending on "Star Wars" is less than the cost of Apollo, ISS, or even NASA's fleet of space telescopes.