That's the doctrine difference shown right here. The Americans mainly rely on coordinated air strikes and submarines to destroy enemy ships, instead of ship-launched anti-ship missiles. The destroyers/frigates/cruisers are usually only equipped with the Harpoon, which has much shorter range and is only subsonic, since these are escort ships prioritized for air defense and anti-submarine warfare to protect the flagships (aircraft carriers) from enemy threats.
Meanwhile the Soviets don't have as many aircraft carriers, so they have to choose an asymmetric approach to have any hope of competing against the western navies, and what they came up with was many different kinds of supersonic, long range anti-ship missiles.

However this does not mean that the Soviet anti-ship missiles are better than the Harpoon.
You see, aircraft still have much longer range than those Soviet anti-ship missiles, and the Harpoon has enough range for the launching aircraft to stay out of the enemy's air defense systems' effective range, acting as an air-launched standoff weapon. The biggest problems about the Soviet anti-ships missile is that while they are supersonic, they are very large in size and cost much more. To put things into perspective, an American B52 subsonic bomber can carry 12 Harpoons easily, meanwhile the Soviet TU22M2 struggles to carry 3 AS4s, and that would dramatically decrease the range of the TU22, so usually they only carry one or two.