Look at their understandings of the history of interaction between the US, in particular, and the Islamic world broadly. Obama makes a push to move on from that history -- to reset (or "overcharge"?):
The relationship between Islam and the West includes centuries of coexistence and cooperation, but also conflict and religious wars. More recently, tension has been fed by colonialism that denied rights and opportunities to many Muslims, and a Cold War in which Muslim-majority countries were too often treated as proxies without regard to their own aspirations. Moreover, the sweeping change brought by modernity and globalization led many Muslims to view the West as hostile to the traditions of Islam.
Fair. Now Here's Bush, way back in 2003 (the full speech has been scrubbed from the White House page, so I'm linking to the excerpts I can find):
Sixty years of Western nations excusing and accommodating the lack of freedom in the Middle East did nothing to make us safe -- because in the long run, stability cannot be purchased at the expense of liberty. As long as the Middle East remains a place where freedom does not flourish, it will remain a place of stagnation, resentment, and violence ready for export. And with the spread of weapons that can bring catastrophic harm to our country and to our friends, it would be reckless to accept the status quo.
Therefore, the United States has adopted a new policy, a forward strategy of freedom in the Middle East. This strategy requires the same persistence and energy and idealism we have shown before. And it will yield the same results. As in Europe, as in Asia, as in every region of the world, the advance of freedom leads to peace.
The advance of freedom is the calling of our time; it is the calling of our country.
In fact, Bush was harsher on Islam's critics, in an important way, than Obama. Here's O:
The attacks of September 11, 2001 and the continued efforts of these extremists to engage in violence against civilians has led some in my country to view Islam as inevitably hostile not only to America and Western countries, but also to human rights. All this has bred more fear and more mistrust.
Here's Bush in 2003:
Some skeptics of democracy assert that the traditions of Islam are inhospitable to the representative government. This "cultural condescension," as Ronald Reagan termed it, has a long history. After the Japanese surrender in 1945, a so-called Japan expert asserted that democracy in that former empire would "never work." Another observer declared the prospects for democracy in post-Hitler Germany are, and I quote, "most uncertain at best" -- he made that claim in 1957. Seventy-four years ago, The Sunday London Times declared nine-tenths of the population of India to be "illiterates not caring a fig for politics." Yet when Indian democracy was imperiled in the 1970s, the Indian people showed their commitment to liberty in a national referendum that saved their form of government.
Time after time, observers have questioned whether this country, or that people, or this group, are "ready" for democracy -- as if freedom were a prize you win for meeting our own Western standards of progress. In fact, the daily work of democracy itself is the path of progress. It teaches cooperation, the free exchange of ideas, and the peaceful resolution of differences. As men and women are showing, from Bangladesh to Botswana, to Mongolia, it is the practice of democracy that makes a nation ready for democracy, and every nation can start on this path.
It should be clear to all that Islam -- the faith of one-fifth of humanity -- is consistent with democratic rule. Democratic progress is found in many predominantly Muslim countries -- in Turkey and Indonesia, and Senegal and Albania, Niger and Sierra Leone. Muslim men and women are good citizens of India and South Africa, of the nations of Western Europe, and of the United States of America.
I'd score that point in Bush's column, too.
Obama did hit one good note that I never noticed Bush hitting -- that of America welcoming its Muslims domestically -- and that's a good thing. Somehow, the Wall Street Journal thinks it was a "rhetorical shift on democracy," but really, it's just Bush-lite, filtered through Obama's lense of personal story-telling.
Added: this is also quite worth your time.