I'd be interested in the differences. I know in the 2000 election, Gov. Bush stated that he was standing on the ground of being a compassionate conservative. Was this part of the differences?
Bush was much more on the Evangelical wing of the party due to his Born Again experience that turned his life around (he was a chronic alcoholic beforehand), and his ideology therefore was a bit more socially conservative (Cheney was more open to gay marriage, while Bush saw it as strictly between male and female), as well as a bit more economically centrist. Bush was known for saying things like "Family values do not end at the Rio Grande" or "I don't believe in balancing the budget on the backs of the poor".
Cheney was more of a behind the scenes guy, did not care much for social issues, seeing them as a distraction, but economically was much more cronyist than Bush. He was also much stricter on the border.
Bush was probably also much more idealistic in terms of foreign policy than Cheney. Bush's anger with Cheney (and Rumsfeld, for that matter) led him in his second term to go all in with Condy Rice's Freedom Diplomatic Doctrine, and prevent the Israelis from bombing Iran in 2007, which Cheney disagreed with, wanting the Israelis to go ahead.