I think you'd still see tax cuts, but they'd be a lot smaller and you'd have a much smaller deficit during the Bush administration.
Under Bush, foreign policy would be just as hawkish as under Reagan, if not more so. I'd expect that the invasion of Grenada, the strike on Libya, and the operations against Iran still happen, and the invasion of Panama might come earlier.
I've heard some people describe Bush in 1980 as a pro-choice candidate(?), but I've never seen anything on the matter from Bush the Elder himself. In general, with Bush being much more socially moderate than Reagan, the Religious Right wouldn't come to roost in the Republican Party, instead either shifting between the parties/candidates as seems to have happened in the past(?) or perhaps not even becoming a major political force at all.
Returning to fiscal issues, I'd imagine that Bush keeps Volcker on board just as Reagan did, so the economy recovers roughly as it did OTL, in time for the 1984 election. Where things get interesting, IMO, is what Bush decides to do with Volcker as his term goes on. In OTL, Reagan replaced Volcker with supply-sider Alan Greenspan in 1987. Bush, as I'm sure you know, was no fan of supply-side economics, hence there being smaller tax cuts. Would Bush keep Volcker on board to the end of his administration, or would he replace him? Either way, I'd imagine that supply-side economics are nowhere near as popular among the establishment/mainstream of the Republican Party, so that the GOP establishment stays closer to the legacy of Eisenhower on fiscal matters. I'd imagine that the Ted Cruz-type of ideological, movement conservatives would still advocate supply-side economics, though.
I'd like to see Kemp as Bush's VP. Kemp was a supply-sider, yes, but if Kemp runs as Bush's successor in 1988, then I imagine that he'd have to run closer to Bush, just as Bush in OTL's 1988 election ran as a continuation of Reagan.