Maryland Gov. Harry Hughes met with Baltimore Mayor William Donald Schaefer one day last fall in the mayor's ceremonial office, an ornate room dominated by 10-foot mirrors, to discuss a plan to locate a hazardous waste dump in the city.

The governor's staff person had just begun to explain the plan when suddenly Schaefer swiveled in his chair and turned his back on the presentation. He began tapping his desk impatiently and then exploded: "Jesus Christ, Jesus Christ, Harry. I can't stand this. You don't do anything for me when I need it and now you want me to do this and take this s--- for you."

For the next minute, according to one observer, as the unflappable governor sat silenty, a red-faced Schaefer unleashed a diatribe at Hughes that was a mix of vulgarities and accusations. Finally Schaefer calmed down, agreed to the outlines of the plan and strode out.

Later, Hughes told friends that he was through trying to deal with Schaefer: "I've tried, I've tried, I've tried and I'm not going to try anymore. I'll be fair to the city but this man is irrational." And Schaefer told his associates that he, too, had had enough: "I can't work with him. He just doesn't understand the city."

No one is quite sure when it all began, this iciness between the Mayor of Baltimore and the Governor of Maryland, but as tales of private explosions and public snubs have surfaced, everyone has known it is there. "Somewhere along the line they rubbed each other the wrong way," said Frank Gunther, a Baltimore businessman who has worked for both men. "And it just never rubbed off."

Most people who know them say the friction results more from a difference in leadership styles than over issues, and that although initially after Hughes' upset election in 1978 the mayor and the governor got along, trouble between them was inevitable.

The balding Schaefer, a bachelor who still lives with his mother in the row house he grew up in, likes nothing better than to comb the streets and alleys of his city, shaking hands and listening to residents' problems. He has become Maryland's best ham to promote Baltimore--wearing a striped bathing suit to jump in the seal tank at the new aquarium, literally singing its praises on television advertisments. Schaefer got his start with the Democratic organization that spawned former governor Marvin Mandel, and his relations with Mandel were extremely warm. His style of running the city, and his hot-tempered impatience, has come to be known as "governing by temper tantrum." His idea of a vacation is a trip to his Ocean City trailer where he can fish off the dock.

Hughes, a native of rural Denton on the Eastern Shore, is the perfect media candidate, tall and distinguished, but reserved and somewhat uncomfortable with handshaking and back-slapping. Hughes wouldn't be caught dead in a striped bathing suit. He made his way to the governor's mansion as a man of integrity, implicitly campaigning against everything Mandel stood for. He is a lawyer by profession and temperament and can spend months picking over a problem before reaching a decision. His personal friends are judges, lawyers and businessmen. After his 1978 upset election victory, he bypassed Maryland's beaches for Jamaica.

"If you tell Donald Schaefer you have a pothole," said Ann Lewis, political director of the Democratic National Committee who has worked with both men in Maryland, "he'll be back to you in two days and either have it fixed or find out why. If you tell Harry Hughes, he'll commiserate with you."

Said one Balitmore politician: "Harry Hughes, who I am credited with getting along with and liking, is one of the most enigmatic, uncommunicative people I have ever seen in public office. Under the best of circumstances Hughes is not an easy person. How do you talk to him? Most politicians, even preppy politicians, have some blood. Donald Schaefer just can't deal with it."

Because this is a gubernatorial election year, the prickly relationship between Hughes, the most important politician in Maryland, and Schaefer, who may be the second most important one, has become more than an item for gossip.

Baltimore is the largest subdivision in the state--the hard-pressed port town is Maryland's only real city--and produces about 20 percent of the state's votes. While Schaefer is far from being like the organization bosses who controlled Baltimore 20 years ago, he is an extremely popular man.

It is not surprising, then, that Hughes and his supporters have taken to minimizing the rift, while the governor's opponents delight in pointing it out, chuckling when Schaefer ignored the opening of the Hughes campaign headquarters, called Republican gubernatorial candidate Robert A. Pascal "my friend" and wished Hughes' Democratic opponent, Baltimore State Sen. Harry J. McGuirk, "good luck" in his quest.

Schaefer, for his part, says that Hughes is a "very honest man" with whom he wanted to work. But he says Hughes has not properly responded to his pleas for help for his city. He felt snubbed by the lack of immediate response to his hand-written messages to the governor and has felt that the governor treated Baltimore just like every other subdivision when it faces many more problems.

Hughes took too long, he said, to respond to the city's trash crisis, has not set up a jobs program to help with its overwhelming unemployment problem and has not voiced disapproval over a suit seeking to overturn the state's system of funding education that in effect subsidizes Baltimore.

Hughes has kept his personal feelings about the mayor mostly to himself. Recently he acknowledged that the mayor's public snubs and private outbursts have been a little disconcerting. "I've been amazed at times by his Schaefer's reactions."

The governor bridles at the suggestion that he has not been responsive to Baltimore, citing a record stretching back to his days as a state senator from rural Caroline County. "I think the record shows we've given more to Baltimore than any other subdivision," he said. He mentions increases in police, education and port fire protection aid and his decision to pick up the cost of several programs cut by the Reagan administration that affected Baltimore more than any other jursidiction. He also pointed out that despite federal cutbacks, the city has not had to raise its tax rate, in large part because of state largesse.

Even those around Schaefer agree that, given events at the federal level and the state's own fiscal troubles, the city has done well under Hughes and point out that on many major city issues, such as the state bond issue to renovate Memorial Stadium, the two men have been able to work together.

But they say that the current tiff is not just Schaefer's typical preelection toying with the incumbent (before one past election Schaefer stopped speaking to Mandel and before another publicly embarrassed Acting Gov. Blair Lee III by refusing to shake his hand).

At first things were quite friendly, according to both the governor and the mayor. Hughes invited Schaefer and a friend to prime inaugural seats, the two met several times for working breakfasts that the mayor said were "very good, very productive," and Schaefer made at least one foray to the Governor's Mansion.

But within six months the first signs of friction appeared. Schaefer was incensed that Hughes would not respond to his hand-written, staff-delivered notes. A week or more would go by after he sent a message and Hughes would not respond or would turn them over to a staffer. "I don't write that many messages," said Schaefer, still upset by the issue.

Said Hughes: "He doesn't think I communicate with him enough, call him enough, but there is good reason--I've got the whole state. This is a big office. I have 24 subdivisions."

In some cases, Hughes' response was not what Schaefer wanted, feeding a growing feeling of resentment, according to the mayor's associates. Once Hughes invited Schaefer to the Governor's Mansion and Schaefer, having come with a list of city problems to discuss, found that Hughes wanted to read him a press release he was preparing that dealt with city problems. Another time, Schaefer came into Hughes' office demanding $13 million and Hughes responded too hesitantly for Schaefer.

"He listens to me but he doesn't respond," said Schaefer. "He's a total introvert. You tell him your problems, he crosses his arms, head cocked to the side and says nothing. That's okay as long as eventually I get a response. Even a no."

"I don't think anybody does anything fast enough for Schaefer ," responded Hughes. "But there are rules, regulations, laws. You know, dealing with 20 city councilmen all from the same city is very different from dealing with 188 legislators."

The largest crack in the relationship came when the city was forced to close its old, overflowing landfill. Schaefer decided he had to swiftly open a new landfill site even though the city had not obtained the necessary state permits. Hughes said he and his aides contacted Schaefer and told him they would have to take immediate court action against the city, but would also encourage a judge to declare a trash crisis and grant the city the temporary right to operate the landfill. Schaefer agreed to this, according to Hughes and his aides.

When the state went to court, however, Schaefer became enraged. He accused the Hughes administration of being insensitive to Baltimore and uninterested in finding a solution without running to court.

"I was flabbergasted," Hughes said.

So was Schaefer.

"That was the thing that made me the maddest, when he had the attorney general sue me. He had the department head read me the law. We had no place to put garbage. I told him, I wrote him, I told him I advised him in every possible way. I talked to the department head and told him we have trouble and he read me rules. I said, 'I need your help and the governor's help' and he read me the rules."

The situation was resolved as the Hughes people predicted, but the issue did not die for Schaefer. When Hughes journeyed to Baltimore to discuss hazardous waste and Schaefer blew up, it was the suit over the landfill to which participants said Schaefer kept returning.

Now, say associates of the two men, Schaefer has firmly fixed in his mind that he and Hughes will never get along. "He's not going to come around," said Schaefer. "His close people don't like me or my style."

Schaefer so far has avoided publicly endorsing any of Hughes' opponents, either his friend Pascal, the Anne Arundel county executive, or his longtime political associate, McGuirk. While he has let it be known he likes both men and thinks either of them would do a good job, Schaefer also has told friends privately that as a Democrat, he will support the Democratic nominee.

"I would certainly like to have his support," said Hughes. "Because he's a good mayor and he has a following. The thing that disturbs me more than anything is for anyone to think I am not sensitive to the needs of the city and his support would indicate, yes, the governor is sensitive to the needs of the city."