I'd say that our foreign policy is likely similar except without Cheney's influence, things might not go off the rails too much and Saddam would have been handled differently.
Lieberman would have fully encouraged Bush to go after AlQaeda after 9/11. He was one of the more hawkish Democrats, to the point that he eventually became an Independent after breaking with the party.
A Jewish man second from the top probably would not make the Arab World all that happy.
Economically, I think we still get the first tax cut but not the second one, and I think that Bush would likely divert more money to social programs as he stated in his campaign; Cheney was economically not too far from center but also not really buying into the compassionate conservatism thing that Bush espoused and steered him away from that in general.
The two would likely find a way to work with each other and I doubt that there would be too much of an issue. Ideologically, I don't think that they were all that far apart.