The main thing Greedo's shot changes, is Han's competence, and the clarity of the scene, as well as the plausibility, tension and realism of all of the film's danger.
In the original version, Han shoots Greedo once Greedo tells Han that Greedo intends to kill Han.
Han: "Over my dead body."
Greedo: "That's the idea. I've been looking forward to this for a long time."
Han: "I'll bet you have."
(BLAM - Han shoots Greedo)
So the instant Han knows Greedo plans to kill him, he blasts Greedo. Now, if anyone thinks that Han should have waited until Greedo shot and somehow missed at point blank range with a ready blaster that has been aimed at Han the whole time, that person is someone who probably would not have survived this situation if they were in Han's shoes. This change may or may not make Han more moral, but to me it clearly makes him less competent, and less likely to have survived his dangerous lifestyle this far.
As for Han's morality, even in the original version, Han could have shot sooner, but waited until he knew Greedo's intent to kill him. Waiting for Greedo to actually shoot would be more pacifistic (but who ever said Han was a pacifist) and slightly more morally justified, but is also far less competent in terms of Han's chances of staying alive. Saying "Greedo shot first" goes along with "Han should have died", and if Han had died, he wouldn't have been around to save Leia and Luke multiple times, and so a different kind of morality might say that the more moral choice was to blast the murderous Greedo (else who knows how many others Greedo might have gone on to kill later) and live to do more righteous er, smuggling and good deeds another day.
It also undermines the clarity of the scene, because how did Greedo manage to miss? Is this now going to become a universe where moral Midichloridians intervene to spoil the aim of bad guys? Could be that IS how Lucas thinks, and would explain the Stormtrooper inaccuracy phenomenon, and especially the ludicrous prequel survivals of Anakin and Jar Jar in the many scenes where they would by odds have died over and over and over. By having Han shoot first, it removes the need for unbelievable badguy failure and puts responsibility for Han's survival on his own trigger finger.
I say this undermines the plausibility, tension and realism of all of the film's danger. In the original, Han seemed to be in serious danger of being shot by Greedo unless he shot first. In the Greedo Shot First version, having a bounty hunter sitting across a table from you aiming a loaded blaster at you doesn't seem to be all that big a threat, because you can heroically flinch to make him miss and then shoot him, and count on him not dodging your shot. This makes no sense, is unbelievable, and changes the nature of the film's violence from a situation where good people can and regularly do get shot and killed, to one where extremely dangerous situations never kill the good guys and don't even need any kind of good reason why they didn't.
And, more importantly, are there any points anywhere in Star Wars
canon that depend on this change? For example, are there stories where
Greedo's friends or family seek revenge because he was gunned down in
cold blood that were effected because of this change (so Han's shot
was in self defense) or maybe stories of Han Solo's earlier life that
make him less of a scoundrel than we would consider him if the footage
of that scene was not altered?
Well there's Greedo's employer, yes. See the next film, The Empire Strikes Back.