To what extent did the two administrations of George W. Bush both follow and depart from the Reagan legacy?

Certainly the Bush Administration sought to claim the legacy of Reagan as its own, and many of Bush's domestic policies shared an ideological affinity with those of Reagan. Like Reagan, Bush believed that the key to economic growth was through what have become known as "supply-side" economics, an approach rooted in cutting taxes. Like Reagan had, Bush sought fundamental reform for the nation's tax code, slashing more than one trillion dollars in income taxes in the first two years of his presidency. Reagan's tax cuts were primarily aimed at the nation's highest earners, who saw their tax rates tumble by almost half during his presidency. Like Reagan, Bush then faced significant budget deficits and an increase in the federal debt, and he declined to raise taxes in response (as Reagan had, mostly by closing "loopholes" in the tax code.) Bush was, perhaps even more than Reagan, a social conservative. He advocated limited privatization of Social Security, opposed abortion, and promoted federal funding for faith-based civic service. On the other hand, Bush oversaw a significant expansion of the federal government's role in education. His No Child Left Behind Act was one of the centerpieces of his agenda, and it involved the federal government in education, a power traditionally reserved to the states, in unprecedented ways. 

In terms of foreign policy, the challenges each President faced to a large extent dictated their actions. Reagan was an ardent Cold Warrior, and his presidency marked an escalation of this conflict, albeit also one that emphasized dialogue with the Soviet Union. Reagan authorized military and economic aid to anti-Communist fighters in Central America, Afghanistan, and elsewhere. He increased military spending to unprecedented levels. Above all, he sought to frame the Cold War as an ideological conflict. George W. Bush, of course, had to contend with the aftermath of the attacks on 9/11, and his "War on Terror" was in many ways undergirded by the same assumptions about the supremacy of western democratic ideals as those espoused by Reagan before. Many of Bush's advisers had first risen to prominence under Reagan, and it is impossible not to hear echoes of Reagan's "evil empire" speech referring to the Soviet Union in Bush's "axis of evil" reference to Saddam's Iraq, Iran, and North Korea. Bush and his advisers, like Reagan, fervently believed that the United States had a leadership role to play in what both viewed as the "free world." His invasion of Iraq to achieve regime change was further than Reagan ever went, but certainly both presidents believed in an activist, aggressive foreign policy based on an unwavering faith in American ideals. 

Ultimately, it is probably correct to say that Reagan didn't live up to the Reagan legacy, which has become more of a political discourse than an accurate reflection of his policies while president. Reagan was, for instance, fairly moderate on such issues as immigration (where Bush, too, took a centrist course) and gun control (which was becoming a major issue for the right at the end of Bush's presidency). Both men shared a vision of America that drew on the same themes of national honor, limited government, and American exceptionalism, though they, like all presidents, were forced to abandon or deviate from this ideology in response to the real-world challenges of the presidency.