Yeah we're not given any indication at all for why Shinzon has it in so badly for Earth. About the most we can surmise is that it comes from jealousy over Picard's perfect life, and a desire to make Earth pay for not rescuing him or something.
But the movie never really suggests or explores any of that.
Watching the special features on the dvd, the creative team used themes from the last Arthurian legend. Notably the Battle of Camlann, where Arthur's son Mordred kills him.
With Picard=Arthur and Shinzon=Mordred.
Arthur neither Aruthur or Picard have any children to continue their lines. Both are great and renowned warriors.
Arthur was initally happy to find out Mordred was his son. He felt he could be the worth heir to all that Arthur built. Similarly Picard put alot of trust and hope in Shinzon. With Shinzon's talks of peace between the Federation and Romulus. Picard probably felt this was a chance to have a part of himself continue doing great things.
Mordred betrayed Arthur and sought to destroy everything Aruthur stood and fought for and kill Arthur. Shinzon withheld his true intentions from Picard. Until he was caught with a WMD biogenics weapon, and later tried to kill Picard 3x.
Mordred and Shinzon's vendettas against their "fathers" are personal. Mordred is young and want to forge his own kingdom of war, but first has to snuff out his old man. Alot of people think Shinzon parallels Khan, but not really. That title goes to the holo-clone character that is Nero of ST09. Compared to Picard, Shinzon went from being a promising infiltration clone, to slave, to shock trooper in the Dominion war, to successful military commander, to Praetor of the entire Romulan Star Empire. Picard has done a lion share of great things, but he really only captains a ship in the larger scale of the galaxy. In the film Shinzon does voice his hatred that he is only a shadow, or echo of Picard and that his life has no meaning while Picard still exists. The plan was always to decimate Earth, but for Shinzon it became a pressing priority. If he succeed, then Picard, and everything he stood for would be figuratively null and void. There would be no Federation or history record of Picard's deeds of beliefs. Shinzon did say that his name would echo through eternity while Picards would fade into a dim memory.
Lastly Shinzon and Mordred's deaths are the same. Arthur impales his son on a lance, and Picard kills his clone with a lance like object. In the Battle of Camlann, Arthur too was mortally struck and died of his injuries. I am one of the few people who think it would've been better for Picard to die and not Data. While Data's sacrifice was as noble as Spock's in TWOK. I think they could've done a better job in making it special. I think we the fans and general audiences forget that Data is an android who is learning to be human. The theme of an artificial life form willing committing and acknowleding the concept of self sacrifice is powerful. Brent Spiner in interviews said that was his intentions with Data's death, but people missed point.
If Picard had died then the crew would'nt have split up at the end of the movie. Riker would've been promoted to captain of the Ent-E since he was technically still commander of that ship. Data and Worf would've been promoted. Beverly would've likely stayed on, to the family together and such as the crew would fondly remember Picard and all the times they shared.
Spelling it out like this the movie could have been ZOMG amazeballs story wise. But Stuart Baird wasn't hired to direct a thought provoking drama, he was hired to direct and action movie. All the themes i brought up are in the movie but you would have to know what John Logan and Spiner were aiming for to fully get the message.