Washington (CNN) -- Nothing, it seems, can bridge the bottomless political divide in Washington over taxes and spending.
Not an election last November that gave President Barack Obama a second term. Not polling that shows a strong majority of Americans want both sides to compromise in forging an agreement to reduce chronic federal deficits and debts.
Not the president's new personal outreach to Congress, including a 90-minute meeting Wednesday with House Republicans. And not even white smoke from the Vatican chimney that signaled selection of a new pope as the talks occurred.
"You are straining the analogy," Obama told reporters afterward when asked if the meeting produced any similar message of spiritual significance.
Leaders on both sides acknowledged the deep differences between them on a day when congressional committees began considering separate spending proposals for 2014 to launch the formal budget process.
In an interview broadcast Wednesday on ABC, Obama warned that compromise may prove unattainable.
"Right now what I'm trying to do is create an atmosphere where Democrats and Republicans can go ahead and get together and try to get something done," he said of his so-called charm offensive that has included a dinner with GOP senators, lunch with House leaders and meetings with others legislators. "But ultimately it may be that the differences are just too wide."
House Speaker John Boehner labeled Obama's visit to the Capitol on Wednesday "productive" because it allowed both sides to understand the "very real differences" between the parties.
"Republicans want to balance the budget. The president doesn't," Boehner said. "Republicans want to solve our long-term debt problem. The president doesn't. We want to unlock our energy resources to put more Americans back to work. The president doesn't."
He then came to either his punch line or understated conclusion: "But having said that, today was a good start, and I hope that these kinds of discussions can continue."
Earlier, House Budget Committee Chairman Paul Ryan also recognized the challenge, telling the panel's first meeting on his proposed budget for next year that Congress faced a tough negotiation.
"We are hoping that at the end of the day, we're still talking to each other and we can make a down payment" on balancing the federal budget, said Ryan, R-Wisconsin.
On the Senate side, Budget Committee Chairwoman Patty Murray made her own joking reference to the papal decision as the panel began considering the first full Democratic spending plan in recent years.
"I understand we have a new pope and a committee hearing to mark up a budget. That's history twice, so it's good," said Murray, D-Washington.
Turning serious, she added that "this process is not going to be easy" because of "a serious difference of opinion about what our government should be doing to keep our economy and our national finances moving in the right direction."
The proposal by Senate Democrats called for a mix of increased tax revenue and spending cuts to reduce deficits by about $1.9 trillion over 10 years.
It would increase revenue by about $975 billion by eliminating and curtailing tax breaks and loopholes for wealthy Americans and corporations. It would also cut spending by an estimated $975 billion: $493 billion in domestic spending; $240 billion in defense spending; and $242 billion in interest savings.
The proposal included a $100 billion economic stimulus package for road and bridge repairs, as well as worker training, that Murray said would be paid for by curtailing tax breaks for high-income households and corporations.
However, the Senate plan avoided significant changes to popular entitlement programs such as Medicare and Medicaid, which are major drivers of federal deficits.
Republican leaders oppose new tax hikes or revenue and demand substantial overhauls of entitlement programs, setting up another in the litany of congressional impasses of recent years.
At the White House, spokesman Jay Carney told reporters that Obama's new outreach was intended to find "common ground" on a deficit reduction plan that would include reforms of the tax system and popular entitlement programs.
Obama will introduce his own budget proposal next month, and the president and Democrats concede their approaches would not eliminate annual deficits, as sought by Republicans, but instead reduce them to what they say are manageable levels.
Republicans call such an approach inadequate, insisting that government has become too large and costly to ensure needed economic growth.
Ryan proposed a conservative budget for fiscal year 2014, which begins on October 1. He said it would eliminate the annual deficit in a decade without raising taxes.
It calls for cutting $5 trillion from projected spending increases in the next 10 years while lowering tax rates and getting rid of most of Obama's signature legislation of his first term -- the 2010 health care reform law.
Ryan also revived his proposal to reform Medicare, the health care program for senior citizens that is considered the biggest driver of rising federal deficits as costs increase and more Americans become eligible.
The idea was a major issue in last year's presidential election, in which Ryan was the vice presidential candidate on the GOP ticket that lost to Obama.
It calls for offering senior citizens a choice between traditional fee-for-service Medicare and a premium support system that would provide a fixed government payment to help them buy private health insurance. The plan would take effect in 2024 to exempt people 55 and older today.
Both sides reverted to harsh rhetoric from last year's election campaign in defending their deeply entrenched positions on Wednesday.
Ryan referred to "job-killing tax increases" pushed by Obama and Democrats, saying his push for less government and lower tax rates would benefit economic growth.
"What we're saying, like we said before, is we can hit these same revenue numbers without killing jobs," he contended, adding that budgets were all about priorities and choices.
Democrats responded that the Ryan proposal would harm economic growth by shifting the burden of deficit reduction to middle-class Americans, the elderly and others, while cutting spending for college loans, infrastructure development, scientific research and other areas vital for job creation.
"It is not consistent with American values. It is not fiscally responsible," argued Rep. Kathy Castor, D-Florida.
In the ABC interview, Obama took aim at the choices in Ryan's budget plan.
"If you look at what Paul Ryan does to balance the budget, it means that you have to 'voucherize' Medicare, you have to slash deeply into programs like Medicaid, you've essentially got to either tax middle-class families a lot higher than you currently are or you can't lower rates the way he's promised," the president said.
For his part, Ryan noted that while his side lost last year's election, his 2014 budget plan adhered to GOP principles that the party believes will set the country on the right path to growth and prosperity.
However, he sounded contradictory when challenged by reporters Tuesday about his call to eliminate most of Obama's health care reform law. After first saying "we are not going to re-fight the past" and that "law is law" with regard to the measure known as Obamacare, Ryan later said that "we need to repeal and replace Obamacare with a better system."
Republicans led by their conservative base seek to shrink the size and cost of government, opposing any new tax revenue while pushing for spending cuts and lower tax rates that they say will spur more economic growth.
After agreeing in January to allow tax rates on top income earners to return to higher levels of the 1990s, Republican leaders say they oppose any further steps to raise taxes.
Obama and Democrats say they want to protect Medicare, Medicaid and Social Security, and that comprehensive deficit reduction must include increased taxes on wealthy Americans to prevent the burden of austerity steps from shifting too much to the middle class, the elderly and other vulnerable demographics.
By clearly staking out positions in their budget proposals, Obama and Congress appear intent on trying to avoid the crisis-driven brinksmanship of the past four years.
However, the familiar, partisan nature of the budget plans illuminated the continuing political division that the public blames for legislative dysfunction.
A CBS News poll last week showed more than 70% of respondents want both sides to compromise to end the brinksmanship over taxes and spending that dominated Obama's first term.
During the past four years, House Republicans pushed through partisan budgets that Senate Democrats ignored, forcing the repeated extension of past spending plans.
Meanwhile, the president's budget proposals generated little support in Congress.
The upcoming negotiations are complicated by lingering fiscal issues from past showdowns.
Deep cuts to military and other discretionary spending took effect this month, and both sides were expected to try to soften their impact through a separate funding measure for the rest of the current fiscal year, which ends September 30.
Called a continuing resolution, it must pass by March 27 to prevent a partial government shutdown. The Republican-led House passed its version last week, and the Democrat-led Senate took up its own version this week.
Congress also must authorize an increase in the federal borrowing limit this summer, and Republicans have made clear they intend to leverage that moment to try to extract concessions.
A comprehensive deficit-reduction deal appeared close during Obama's first term, but eventually fell apart over taxes.
Obama and Democrats want to eliminate tax breaks and loopholes worth about $600 billion over 10 years as part of a broader $1.2 trillion deficit-reduction package that would include entitlement reforms.
Some Republicans have indicated support for ending such tax breaks as part of a broad deal. However, the fiscal-cliff agreement in January that resulted in higher tax rates on top income earners galvanized opposition by GOP leaders to further increases in tax revenue.
Meanwhile, Republicans say Obama and Democrats must deliver on significant entitlement reforms.
One change Obama has proposed would tighten the adjustment for inflation of benefits such as Social Security, meaning annual increases for future recipients would grow at a slower pace.
Opponents of the reform, known as "chained CPI" in reference to the Consumer Price Index it involves, argue it hurts senior citizens and others who most need their benefits.
If achieved, a grand bargain would give Obama a major political victory and a boost in cementing his desired presidential legacy after the controversial health care and Wall Street reforms of his first term.
Republicans also would get credit from moderates and independents for a willingness to compromise, but conservatives could punish them with primary challenges in 2014 and beyond.
Another possible outcome is a limited agreement that would include some elements under discussion.
For example, a smaller agreement might end some tax breaks and loopholes while cutting Medicare costs paid to providers, not beneficiaries, to achieve $500 billion or so in deficit reduction over 10 years.
Such a result, coupled with previous spending cuts and the January fiscal-cliff deal, would fail to reach the total $4 trillion in deficit reduction over the next decade that economists and political leaders have targeted as the minimum amount needed.
It also would allow both parties to simultaneously claim credit for making some progress while continuing to blame the other for preventing more.
A status quo outcome -- no major deficit reduction steps -- would mean continued brinksmanship over each pending fiscal deadline, as well as further economic uncertainty that already has lowered the U.S. credit rating and slowed growth.
CNNMoney's Jeanne Sahadi and CNN's Ted Barrett, Deirdre Walsh and Ashley Killough contributed to this report.