Two things that would/could be different:
Firstly, Reagan would not be mythologized. His Presidency made into a myth by a very aggressive, very thorough, and very good GOP and Conservative PR operation starting when he left office, and carrying on throughout the 90s. The GOP has since been running on this myth ever since; the idea that Reagan carried out this extremely Conservative program, without compromise, which was a raving success, and which everyone loved and agreed with. In truth, Reagan overreached on many of his policies, such as the tax cuts, which lead him to raise taxes 11 (I think it was 11) times, being the biggest tax raise in history, and he carried out a policy that was moderately Conservative compared to the GOP today, did work with his Democratic rivals (and the Soviets, for that matter). The majority of the nation also disagreed with almost every single policy position of President Reagan and the Conservatives, though most did like him personally, but of those that said they liked him personally, 70% said (pre-1984 election) that they would not be reelecting him. And the reason he did get reelected was that the economy rebounded in time for 1984. And still, most people did not agree with the positions on issues he and the Conservatives had. His administration's approval rating overall was decidedly average, and his administrations actions were quite mixed (deregulation has led to problems which caught up with us in 2008, and continue).
That myth is something very unique, because while other Presidents are idolized, the Reagan myth is so so distant from the actual man and history. And I don't mean that to slander Reagan, but it is what it is, and he is a man of mixed results. And this is something the modern Republican party and Conservative movement have been running on since at least the 90s. Its the gasoline that powers the modern GOP, and says whats ok and what isn't. So without Reagan, and with a moderate like Bush, you will not have that myth, and that seems like it'd create a more moderate GOP.
Secondly, Liberalism could bode better. The reasons include no Reagan myth, a Moderate rather than Conservative administration and national policy in the 1980s, and perhaps the Democrats keeping a better head, and being able to be in a better position in 1988. 1988 was really the last hurrah of Liberalism; Dukakis was beaten badly (though looking promising in some polls in the race), Bush won, and the next Democrat would be a Centrist. The reason the Democrats went more Centrist in time, and were willing to fall in line with more conservative policies than liberal ones if pushed, was because they made the mistake of thinking that Reagan's victories and personal popularity meant the public liked the Conservative ways of the GOP. Which is not true, as I pointed out earlier; the public disagreed across the board, and would agree with the Democrats on their positions on the issues (even on issues you might think the public would be against, such as affirmative action and womens' issues). But, the Democrats got dumb, and turned their backs on that thinking the public was in a place it wasn't. The Democrats fumbled the ball there, and I'd hypothesize that's why the Democrats have had the Bejesus whipped out of them many a time since Reagan left office. So if there is a Moderate in the White House, the Democrats may not go out to the wilderness, and may see fit to stay closer to their tried and true positions. Which I think would do them better than trying to be GOP-Lite.
On a quick aside, if anyone reading this is combating my statements with the assertion "Well, why did Reagan win and Bush win if the public didn't like what they said", its the 4 words that won Clinton the White House: "It's the Economy, Stupid". The public is like a mass in many areas, and one of those is the polls; it is diverse, but it gets happy where good things are and upset from where bad things are. When things are good, approval ratings go up, when they're bad, approval ratings go down. The economy was starting to get better in time for 84, approval ratings go up. Things are ok in time for 1988, Bush wins. The economy tanks under Bush, Clinton wins in 1992.