In Z, it was mainly everyone acting way less retarded thanks to the good group of characters they had around them if you ask me.
The influence of Amuro and Quattro on the Destiny cast was pretty noticeable, especially Quattro on Athrun. Also, Kamille and Setsuko (real robot main character) being Shinn best buddies helped keep him in check and providing him some well needed help.
Best part to me was definitely the way the whole cast (not just Destiny's) looked down on the Archangel : for some of them, they were plain morons who didn't knew what they were doing. Which they always, but without anyone to tell them in the series. Their arrival in the group near the end is pretty rough at first, especially some parts like Shinn and Mwu meeting. It's SRW so of course everyone ends up playing nice though.
Also, if you fulfill certain conditions or are on your second playthrough, you can get Shinn to actually act on his own and decide to stay on the good guys side, which triggers a totally different endgame. Luna stays with him but the Minerva goes back to ZAFT as planned, and Ray with them, cursing Shinn's betrayal.
Dunno about the "normal" route, but on the one Shinn's stays with you, Destiny's final stage is merged with Zeta movie's. Notable change is Haman siding with you on that route, so no Neo Zeon in the mix.
There aren't that much change if you ask me, it's pretty standard SRW changes, but it automatically works better, thanks to a better environment. Like how Evangelion cast was more likable in SRW@ thanks to the interactions with other characters IMO.