WASHINGTON — When it comes to resolving their “fiscal cliff” impasse, the dollar gap between President Barack Obama and House Speaker John Boehner is tiny in federal terms. That masks a monumental political ravine the two men must try to bridge, with most of the burden on the now-beleaguered Boehner.

Short of support from his own Republican Party, a chagrined speaker abruptly canceled a House vote Thursday night on his so-called Plan B. The measure would have prevented looming tax increases on everyone but people earning over $1 million annually, but was opposed by rank-and-file Republican lawmakers unwilling to vote for any tax increases at all.

Now Boehner, Obama and Senate leaders face fresh bargaining over a broad package of tax increases and spending cuts, with Thursday night’s GOP retreat weakening Boehner’s leverage. Ticking ever louder is the start of the new year, which by law will usher in hundreds of billions in tax increases and spending cuts — the “fiscal cliff” — unless leaders avert it by crafting a compromise deficit-cutting package that can get through the GOP-run House and Democratic-led Senate.

“I’m interested in facing the major problems that face our country,” Boehner, R-Ohio, said Friday, calling for the White House and congressional leaders to work together. He added later, “How we get there, God only knows.”

Despite the impassioned political clash that the “cliff” has prompted, weeks of intermittent bargaining between Obama and Boehner have left them facing relatively miniscule dollar differences by Washington standards.

Obama wants to raise taxes by about $20 billion a year more than Boehner. The two men differ over spending cuts by roughly the same amount.

By almost any measure, $20 billion is real money. Yet compared to the $2.6 trillion the government expects to collect next year and to the $3.6 trillion it plans to spend, $20 billion barely registers — less than 1 percent of what the government already is on track to raise and spend. Relative to the U.S. economy, which should weigh in at well over $15 trillion next year, $20 billion is even smaller.

“The policy implication is very slight,” Robert Bixby, executive director of the Concord Coalition, a non-partisan anti-deficit group, said of the $20 billion gaps between Obama and Boehner. “It’s not worth the price of not getting a deal. And the impact on the economy is totally insignificant.”

On the other hand, economists have warned that the “cliff’s” massive tax boosts and budget cuts would heave the economy back into a recession, although likely a brief one.

Though the numbers separating them are small, Obama and Boehner have real policy disputes. Yet their inability to strike a compromise so far underscores that their problem is more than arithmetic: It’s largely driven by the difficult politics that Obama and Boehner face in firming up support from their own parties.

Boehner’s clout was weakened by the Plan B debacle, and it remains unclear how many GOP votes he could deliver for any compromise he might reach with Obama. Yet while his Plan B would have received virtually no Democratic votes, a bipartisan accord with Obama likely would get significant backing from House Democrats, lightening Boehner’s load.

Even before Thursday, the president and the speaker each faced formidable political challenges.

Chastened by Obama’s re-election, Boehner has violated a quarter-century of Republican dogma by offering to raise taxes, including boosting income tax rates on earnings exceeding $1 million annually.

Eager for a budget deal that would bolster his legacy and let him address other issues, Obama would cut the growth of Social Security benefits, usually off-limits to Democrats. He also would impose tax increases on a broader swath of people than millionaires — those with incomes over $400,000. That figure is a retreat from what he campaigned on: a $200,000 income ceiling on individuals and $250,000 on couples.

Those concessions mean that both men have angered lawmakers and staunch supporters of their respective parties. Neither wants to risk his political capital by embracing a deal his own party rejects.

“When you walk into a room and represent a group and you have to give ground to get a deal, you have to stay in that room as long as you can and you have to walk out with blood on your brow,” said Joseph Minarik, research director for the Committee for Economic Development and a veteran of grueling budget talks as a former Clinton White House and House Democratic aide. “Otherwise, the people outside the room don’t believe you’ve fought hard for them.”

In their talks, Obama has proposed raising taxes by $1.2 trillion over the coming decade by boosting the current top 35 percent rate to 39.6 percent for income over $400,000, plus other increases on the highest-earning Americans.

He also says he’s offered about $1.2 trillion in spending cuts over 10 years, including slowing the growth of benefits from Social Security and other programs. His proposed spending cuts also include $400 billion in savings from Medicare and Medicaid, the health care programs for the elderly and poor whose defense Democrats consider precious priorities.

Boehner has offered about $1 trillion in tax increases and roughly the same amount in spending savings. An earlier Boehner offer included $600 billion in Medicare and Medicaid savings — well more than Obama — but it’s unclear whether the speaker is still seeking that figure.

Because of a dispute over how some savings are classified, Boehner says Obama’s offer is really $1.3 trillion in higher taxes and only about $850 billion in spending cuts.

The House speaker says Obama’s offer is not balanced because its new taxes and spending cuts are unequal. And he complains it does too little to control fast-growing benefit programs like Medicare, a chief driver of the federal government’s mushrooming deficits.

Yet while their offers are relatively close, another obstacle they face is that even slight changes in the numbers could force politically significant policy alterations.

Adding, say, another $100 billion to the tax increase over 10 years could mean that people with incomes well below $1 million a year would get a tax increase, something Boehner wants to limit.

On the other hand, adding $100 billion more in spending cuts could mean a deeper hit than Obama wants to Medicare. The president prefers to limit Medicare cuts to the reimbursements that doctors and other health care providers receive, but ever deeper cuts could mean more doctors would be likely to stop treating Medicare patients — an outcome Democrats don’t want.