It also ups the stakes for Obi-Wan and Luke; they've just signed on with a smuggler who shoots a man in cold blood in the very next scene. It's effective because it increases the tension: who is this guy? And just how much trouble might Obi-Wan and Luke be in for hiring him?
Showing Han fire in clear self-defense lessens the impact. It hurts the movie because, well, it's bad drama to have Greedo shoot first.