The Americans spent about 16-years trying to perfect an aerial laser system capable of taking down income ballistic missiles. Russia is now pursuing what it calls space laser counter-measures.
The Russians are after a more modest target. Instead of a laser missile shield, their initial effort is, "making inoperable adversary space vehicles’ sensors and optical electronic systems by directed laser beam impulse."
That seems to indicate that the Russians want to target American communications, command & control and spy orbital systems. Instead of blowing them up with anti-satellite missiles, which creates a host of problems, the Russian approach is to fry them - render them deaf, mute and blind.