First, A Vice President would have to be chosen. In order to shore up Conservative support, and appeal to Reagan's base, I think he would seek to appoint New York Representative Jack Kemp. If that does not work, he can nominate Howard Baker, another relative moderate, who was popular among both Democrats and Republicans.
Regarding his Presidency, there would be no Iran-Contra scandal in my opinion. Reagan had initially been misled (in the opinion of myself and other conservatives) that the arms were being sent to an "anti-terror" and "moderate" faction within Iran, whereas the arms were actually being sent to the Ayatollah's Iran. Bush, through past experiance in the CIA, might have investigated further and may or may not have continued the deal; after all, it was in return for the freedom of captive Americans, there is substantial pressure there. If he doesn't, and more direct action is taken to free them, then there likely will be bloodshed in some instances. However, it will in the long run prove to be unpopular in wringing concessions from major powers, and will be done less frequently (hopefully). Of course, this also means that the Contras in Nicaragua lose their funding and collapse.
As others have stated, the militias in Afghanistan will likely have been given additional arms and support, though it may be limited to those leaders (like Ahmad Shad Massoud) who were political moderates. Depending on how much more of a cost this would bring upon the Soviet Union, the war may end sooner, and the moderate possibly could take power in Afghanistan.
Besides these points, I am not sure if his Presidency would be all that different then Reagan's.