Many of the differences would be driven by the ways that the world has changed since 2000. Additional differences between the two Presidencies will be defined not by any differences in political philosophy or goals, but by developments out of their control.
Like Obama, Bush came to Washington promising to find a way to build bipartisan consensus with Democrats. Bush had been very successful in Texas and knew how to work with a legislature, as does Romney. It's hard to predict whether Romney can succeed where Obama has failed in this regard, but he would be likely, like Obama, to be working with a Congress that was friendly for the first two years. He would likely not make Obama's mistake of assuming that the House or Senate would continue to be a friendly one in the third and fourth year of his term. He suggested in the first debate that he would do what he did in Massachusetts and have weekly meetings that including opposition leaders.
Every President has made a transition to leading all Americans across the spectrum of political thought in a way that a campaign doesn't encourage. Candidates must emphasize their differences before taking office, and emphasize our shared goals after election. Reagan really did say during his campaign that he would shut down the new Energy Department, and talked about taking back the Panama Canal, but once in office, he proved to be very effective working with a House of Representatives that had in the previous Congress comprised 277 Democrats (of the 435). Despite the fact that the 1980 elections did see them drop 34 of those seats, the House was run in those days as a big Democratic Party clubhouse where the Republicans were allowed to use the cafeteria. That wouldn't change until 1994.
Obama obviously disappointed his liberal base when he moved to the center after taking office. Bush did the same thing. During the first three sessions of his Presidency, he had a Republican House, and had no excuse for allowing spending to rise as it did, effectively governing as a moderate Democrat might have been expected to. When the Democrats took the house again in 2006, spending really took a dramatic turn north, so there was already a running start when Obama took office. It's impossible to imagine a Bush Presidency without 9/11, Afghanistan, and the intelligence failure that became Iraqi Freedom. We need to remember that the Bush tax cuts were passed and signed in the spring after he took office. The only thing more outrageous than funding a war or two outside the normal budgeting process (which seems to enrage many people even today) would be to do it any other way. How would you feel if Bush had announced in the winter of 2001 that he didn't really want to cut taxes even though we had a surplus, because he was saving up to attack somebody? Finally, Bush had to respond to a financial collapse. The TARP loans show up as increased spending under Bush (and the repayments show up as receipts for Obama). None of these formative events could have been foreseen in 2000.
So President Romney will be sworn in on January 20, and he will have GOP control of both chambers of Congress, as did Bush. But Romney will have a much, much harder job. Bush inherited a healthy economy, with two trillion dollars in tax receipts and a budget surplus. Unemployment was at 4%, and the national debt was around six trillion dollars. By contrast, Romney will start with a single fiscal year 2013 deficit of nearly a trillion dollars (projected, assuming recovery), and unemployment at about 8%, and over sixteen trillion dollars in total debt. Looked at another way, since George Bush took office, receipts have risen 43%, and outlays have risen 113%. This is not something Romney can reverse in even eight years. With deep, painful cuts, he can slow the trend.
Bush was able to give every taxpayer a large tax cut, fight two expensive wars, and deliver a new prescription drug benefit to seniors.
President Romney will have to cut a great deal of spending, and each program has an array of champions. There are not any government programs that are a complete waste of money. (Some can be argued to be short term solutions that create long term problems.) He has laid down a marker, and promised to eliminate government support of popular but comparatively small programs like funding for the Corporation for Public Broadcasting (See the answer from What portion of NPR funding comes from the federal government?). The more significant opportunity to control spending will come by means-testing medicare and Social Security, which will mobilize one of the largest Democratic lobbying groups in the country. "Path to Prosperity" Budget & Plan (2011-12): How is the AARP reacting to Representative Paul Ryan's (R-Wisconsin) Medicare proposals?
Romney has promised to reform the tax system in a way that will be almost equally painful, at least politically. The tax plan should appeal to most people on Quora who are angry about wealthy taxpayers getting away with low tax rates, because it will reduce deductions. It will be revenue neutral by offsetting lower rates with a simpler tax code containing fewer "loopholes." (If I use it, it's a deduction, and if you use it, it's a loophole.) It will put a number of tax advisers and bookkeepers out of business if he is able to accomplish it, which taxpayers should like, but again, every single deduction will have armies of defenders clogging up the halls of Congress, competing for appointment times and floor space with the advocates for all those spending programs under the ax.
To summarize, the differences in these two administrations won't be easily defined by the President himself. Both men are moderates, and both have experience as large state Governors working with legislatures. Neither is famous for being electric as a public figure on television, but both are strong, proven leaders who can meet with fifteen brilliant and strong-willed people, often with different agendas, listen, and then make decisions that will stick. The Congresses of 2001-2008 and the upcoming Congress are affected by these same sobering historical realities. Voters have begun to pay attention to the numbers I cited, and they've dug in their heels. They may want lower taxes, higher services, and an elimination of the deficit. Tough times are ahead for politicians. When service cuts take effect, the mood of the voters might shift again, but in this past election they could not possibly have sent a stronger signal that spending must be cut. Whereas the "Tea Party" brand was effectively weakened over time, the penumbra of voters who never attended a rally but agreed with the concerns they represented has not grown smaller or less engaged.
In addition to other unforeseeable factors, a Romney Presidency could be defined by a new Brie and Chablis Party that marches on the Mall (or possibly a meteor strike that destroys Dallas, or a zombie invasion). I therefore have a modest proposal for Democrats. Their best strategy is to switch (or stay home) and contribute to a big Romney win (and a big Republican pickup in the Senate). The result, if Democrats are right about what Americans want, might very well be an overreaching in Republican-led budget cuts, which would lead to a backlash. If Republicans cut services and programs too deeply, Americans might decide that they do not want to have a government that is so stingy and austere. We have the wealth, after all, as a nation, to provide excellent government services. On the other hand, if the economy thrives when people invest in business instead of T-bills, and people are able to find good jobs, they could leave the new, leaner government in place for a few generations. We've tried ever-expanding government for a couple of centuries. Let's try smaller government. Bush didn't. Romney will. He has no other option.More questions on 2012 U.S. Presidential Election: