To hear former Massachusetts governor Mitt Romney tell it, the problem with President Obama is that he’s incompetent.

Not a bad person. Just bad at his job. A new hire that didn’t work out.

“I think he’s a nice guy. I’m sure he loves the country and wants it to do well,” Romney told a crowd in New Hampshire earlier this month. “I just don’t think he understands the principles that make us who we are.’’

To hear former House speaker Newt Gingrich (Ga.) tell it, however, Obama is actually quite competent. But that’s the problem: Gingrich says Obama is a leftist “radical,” using his considerable political skills to undermine treasured American values.

He’s so good, Gingrich warns, that only another skilled politician can beat him.

By which he means: me.

“Who do you think could go on that platform against Barack Obama and effectively articulate your values, defend your beliefs and communicate his failures without flinching?” Gingrich asked a crowd in Iowa. In case the answer wasn’t clear, he added: “Almost everybody seems to think that I’m a more effective debater than Mitt Romney .”

In their disparate portraits of Obama, the GOP’s two leading candidates have revealed something important about themselves.

Romney is trying to reach a general-election audience, including many people who voted for Obama in 2008 and still like him personally. So he casts the president as an honest mistake, a low performer who simply needs to be replaced.

Gingrich, by contrast, is aiming at a Republican primary electorate that never liked Obama much to begin with. So he portrays the president as the representative of a whole poisoned way of thinking: an adversary who needs to be not just defeated, but repudiated.

Saturday’s results in South Carolina — and Romney’s own recent shifts toward more belligerent language — seem to indicate that Gingrich’s approach might be working better now.

“When Romney attacks Obama, he says, you know, ‘He said he was going to create jobs. And he didn’t. He’s lost jobs. He said he was going to do this, and he didn’t.’ It’s very operational,” said Lynn Vavreck, a professor at UCLA who studies presidential campaigns.

“Gingrich is basically saying that, ‘This is a war for the future of America, between my vision and Obama’s vision,’” Vavreck said. “And that, I think, is a much better message than the Romney message.”

The difference starts, most likely, in the minds of the two candidates. Romney, a longtime chief executive officer, seems to imagine that voters will see this election as a choice of two job applicants.

He opened his campaign, then, with an attack on the other guy’s resume.

“Barack Obama has failed America,” Romney said when he announced his candidacy in June. Then Romney listed all the data that proved it: unemployment still up, home prices still falling, foreclosures still too high.

“Mr. President, you’ve had your chance,” Romney said.

Romney has also attacked Obama for being weak in foreign-policy tussles with Iran and Russia. He also has accused the president for looking to the social-welfare states of European countries for inspiration — trying to turn the United States “into a European-style entitlement society.”

But Romney usually attacks the deeds, not the man. He wouldn’t call Obama a “socialist,” even when Fox News Channel’s Bill O’Reilly asked him to. Twice.

“I prefer to use the term that he’s just over his head,” Romney said in a December interview. O’Reilly pressed him again. “I consider him a big government, liberal Democrat,” Romney said. “I probably am not going to be calling him names so much, as calling him a failure.”

This kind of argument seems designed to play up Romney’s own experience both in business and in the Massachusetts statehouse. And it fits with the buttoned-down air of both the candidate and his campaign.

“The very thing that would make Romney a good president undermines this campaign: being so clinical, and such a detail guy. Nobody wants to know that,” said Frank Luntz, the longtime GOP pollster, who said he is not aligned with either candidate. Voters, Luntz said, “want to know not just what you think — but how you feel. Mitt Romney doesn’t talk about how he feels, because that’s not relevant to what he does.”

In recent weeks, Romney has hardened his rhetoric about Obama: after Saturday’s primary loss in South Carolina, he said Obama wants “to fundamentally transform our country. We want to restore to America the founding principles that made this country great and the hope of the earth.”

But, by then, Romney’s dispassionate take on Obama had already helped open a window for a rival.

“Newt captures the heart,” Luntz said. “Romney captures the brain.”

Gingrich, a veteran of Washington’s message wars, appears to imagine the campaign against Obama quite differently. It is not a personnel decision. Instead, he cast the campaign as the latest battle in a long ideological struggle with the left.

He talked not just about Obama’s resume — but also about his allies.

“His world view is far to the left. The people [he] has appointed [to] the government are very far to the left,” Gingrich told Fox’s Sean Hannity the day he announced his candidacy last year. “And the number one thing we know, from the collapse of the Soviet empire to the failure of socialism in Europe, is left wing policies don’t work.”

Since then, Gingrich’s assessment of Obama’s ideology has become even more strident. In his rhetoric, Obama has gone from being simply “far to the left,” to being “radical,” bent on fulfilling the vision of Saul Alinsky, an obscure and long-dead Chicago activist.

“If Barack Obama can get reelected after this disaster, right, just think how radical he would be in a second term,” Gingrich said after he won the South Carolina primary on Saturday.

But, at the same time that Gingrich has disparaged Obama’s ideology, he has warned of the president’s political skill. In recent weeks, he has argued that Romney is not skillful enough to debate Obama one-on-one.

In this way, Gingrich has been using Romney’s own tactics against him. While the election against Obama may be a cultural battle, Gingrich has cast the Republican primary as a more sober personnel decision.

And in this case, he says, it’s his own résuméthat beats Romney’s.

“He’s not a bad guy,” Gingrich told his Iowa audience back in December. “I’m just saying I’m a more effective debater.”

Staff writer Philip Rucker contributed to this report.