Whether it's a dirty bomb, crashing planes into a building, or some other catastrophic event, you will see a few things happen. Russian nationalism and Putin's power will soar and Chechnya would see a massive retaliation (even if they're not the perpetrators of the attack, a terrorist is a terrorist at this point). Russia garners international sympathy for awhile as well.
Internationally I'm not sure that much would change to the extent that it did OTL. There would certainly be an uptick in pressure on terrorist groups and other countries would be taking steps to make sure it didn't happen to them too. But part of the shock of 9/11 was that the US has rarely faced such brazen attacks. The effect wouldn't be as strong since everyone knows about the Chechens and Russia has had to endure tragedies in even recent history.