Their problems are the same: unemployment, cutbacks, unrest. But they couldn't be more unalike. Doug DeGood, mayor of Toledo, is a rumpled 35-year-old Democrat who lunches on cheeseburgers and chili in a worn industrial city which last year couldn't afford to collect the garbage more than once every two weeks. Margaret Hance, mayor of Phoenix, is a businesslike 59-year-old Republican in a white-collar oasis that has sprung up--from the middle of the desert--to be America's ninth largest city.

As they meet at the U.S. Conference of Mayors, where the talk is of crisis, they both admit things are the worst they've been in years.

"There's an emotional catharsis about this thing, because there's only one of us in each community," DeGood said yesterday, referring to the conference. "It gives you an opportunity to exchange war stories. But I don't find too many people who have a worse war story than having to lay off 40 percent of your municipal workers."

"It's very difficult for people to understand," said Hance, "that I had a pocket of poverty that was 40 square miles--or twice the size of Newark."

The 175 or so mayors, meeting from Wednesday through today at the L'Enfant Plaza Hotel, are mostly Democrats angry with Ronald Reagan and his economic program. As Detroit Mayor Coleman Young, the president of the conference, put it: "People are hungry in this country, and yet he talks about cutting the domestic budget. He's blind to reality."

Which is where Hance and DeGood really split. Hance, whose city has an unemployment rate of 10.5 percent, doesn't blame Reagan. She's sometimes called the administration's "favorite mayor" and was recently rumored to be a candidate for a Cabinet job. DeGood, who has faced layoffs in Toledo's automobile plants, talks of the "insensitivity" and "callous strategy" of the White House.

And, as he says of Hance, "we don't agree on anything."

Toledo is a largely Democratic, blue-collar city of 354,000 on the shores of Lake Erie, a place that DeGood admits is considered "Siberia" by the car company executives who settle there. But DeGood speaks with home-town pride of the symphony, the opera, the zoo and the neighborhoods. He was raised in Toledo, went to its public schools, played high school basketball. He grew up with some of the 13.1 percent unemployed.

"I have had grown men come to me with tears in their eyes," he says. "For the most part, these are people in their forties, people who probably had 12 to 15 years of seniority in a plant somewhere, and who had a great deal of self-pride in their ability to support themselves and their family. Their whole life has gone sour . . . You clearly can't give those people false expectations and say, 'I'm going to see to it that you get a job tomorrow.' I have a feeling of inadequacy. It creates in your mind a guilt feeling that you're not able to remedy that individual's situation."

Things have been bad in Toledo the past few years. AMC Jeep, Chrysler, and the glass companies have laid people off. In the beginning of 1982, DeGood says the city had a $6 million deficit--which is illegal in Ohio. DeGood cut back the garbage collection, closed the swimming pools and stopped cutting grass in the parks. Rats multiplied in the absence of a control program. Potholes grew. From January 1979 to May 1982, almost 1,600 city workers were laid off--representing 40 percent of the city's employes.

"I could feel the wave of resentment and negative reaction welling up in the community," says DeGood. "A good bit of it was directed at me."

Were it not for an increase in the payroll income tax passed last June, Toledo may have defaulted. Now the cut city services are back in operation, and DeGood thinks unemployment may have "bottomed out."

He thinks being mayor is harder than being congressman. "There's no insulation between you and your constituency," he says. "You have such visibility."

He's not sure he'll run again.

Phoenix is a city of 350 square miles and 800,000 people in the Sonora Desert, many of them employes of the electronics and computer industries. For the last decade the Sunbelt has promised warmth, jobs, and the frontier, get-rich-quick spirit of the West. This is Goldwater country, where people are conservative and suspicious of big government.

But last summer, in a nearby county suffering from layoffs in the copper mines, unemployment hit 62 percent.

Hance, who has close ties to the Phoenix business community, sticks with Reagan. "I think it's ridiculous to hold the president accountable for the deficit," she says. "I believe what he's trying to do is right. It's fairly useless to argue about programs unless we're able to get the economy back on its feet. We cannot cure the situation in two years--perhaps not in four."

Hance moved to Phoenix in 1926, before air conditioning. Many of the town's 25,000 residents slept outside in the summer, when temperatures often hit 115 degrees. After a long career as a housewife and community volunteer, she ran for city council 1972. She's now in her fourth term as mayor.

Her answer to the federal budget cuts is to increase productivity and cut back city services. A year ago, Phoenix had its first layoffs in a quarter century. Library and city swimming pool hours have been reduced. The grass in the parks isn't cut as often. And some of the recreation programs have been cut in the schools.

More serious is the condition of the homeless. At last count, more than 300 people were camped in downtown Phoenix on a vacant lot. Sanitation, hunger and shelter are problems.

Says Hance, ruefully, "No one ever thinks there are any problems in the Sunbelt." CAPTION: Picture 1, Mayor Doug DeGood of Toledo: "You clearly can't give those people false expectations . . . I have a feeling of inadequacy."; Mayor Margaret Hance of Phoenix: "I think it's ridiculous to hold the president accountable for the deficit . . . What he's trying to do is right." Photos by Dudley M. Brooks