The eruption of trans-Atlantic acrimony over the U.S. bombing of Libya early this week is rooted in deeply differing perceptions of terrorism between the United States, where the problem is a relatively new and somewhat distant phenomenon, and in Western Europe, whose population has lived with it in one form or another for decades.

Europeans generally feel that the United States, lacking experience with widescale terrorism at home, is naive about the problem, unaware of its complex causes and overly simplistic about how it should be combated.

Mirroring widespread sentiment among American allies who were critical of the U.S. air strike early Tuesday, Italian Prime Minister Bettino Craxi said at a press conference today that "Italy settled accounts with internal terrorism" in the 1970s when Italy fought the Red Brigades, using "patience, firmness and moral force before military force."

Taking umbrage at those in the United States who thought his disagreement with the U.S. strike against Libya represented a lack of courage or national firmness, he said Italy intends "to face any form of terrorism that can involve us or our friends with all the determination and resolve necessary." But, in an apparent allusion to the unilateral U.S. decision, he said that such a confrontation requires that those pursuing the "same objective" work together against the terrorist threat.

In short, many Europeans believe that the United States has much to learn from them about the problem.

"I do not say that we have become used to terrorism, but rather that we have had to learn to live with the pain, the grief and the damage caused by indiscriminate violence and callous groups with political axes to grind," wrote Jonathan Alford, deputy director of London's International Institute for Strategic Studies, in an article published this week by the International Herald Tribune in Paris.

"We have learned, too, how difficult it is to act against terrorism, and there is a sense of weariness when simple solutions are proposed that are intended in one glorious stroke to 'solve the problem.' "

This knowledge is deeply rooted in the European experience of this century through two traumatizing world wars, dozens of bitter civil upheavals, resistance movements, and regional and linguistic conflicts. Among the more recent manifestations of extremist violence are those of the ETA Basque separatists in Spain, the Red Brigades in Italy, the Baader-Meinhof gang in West Germany, the Irish Republican Army in Northern Ireland and a number of groups in France that over the past decades have ranged from Algerian independence fighters to today's extremist Direct Action.

"It is so much easier to talk of terrorism and how to deal with it if you don't have to live with it and cope with it daily as we have had to do here in Europe," said one senior antiterrorist expert in Rome who talked only on condition that his name not be used. "You Americans have never faced real violence in your own territory, within your own society."

This comes from an official who during the 1970s faced the Red Brigades as they "kneecapped" newspaper editors and industrialists or assassinated such symbols of authority as judges, police officials and leading politicians. Aldo Moro, a former prime minister and leader of the majority Christian Democratic Party, was kidnaped in 1978, held in a secret prison for 55 days, then assassinated and left in the trunk of a car in the heart of Rome.

Italian officials take pride in the fact that they faced the destabilizing challenge of Red Brigades terrorism without overreacting with a wave of police repression and without flinching even when it meant refusing to negotiate with the terrorists to save Moro's life. In the end, Italy broke the back of the terrorist movement without compromising its own democratic values. While admitting that there are some major differences between fighting a home-grown terrorist movement such as the Red Brigades and an international group such as those Arab terrorist cells sponsored by Libya's Muammar Qaddafi, Italian officials cited their experience as one the United States should learn from.

West Germany, which faced down an equal challenge in the 1970s from the Baader-Meinhof gang -- whose survivors today call themselves the Red Army Faction -- remains, like Italy, a leading opponent of relying simply on violence to counter terrorism, arguing that overreaction is counterproductive.

At the heart of their argument is the belief that violence per se not only does not halt terrorism, but, in the case of Arab-based terrorism, probably reinforces it and leads to more widespread, and desperate, acts of terrorism. This is what seems to have happened when TWA Flight 840 en route to Athens and the West Berlin discotheque were bombed after the U.S. Navy's first attacks on a Libyan radar installation last month.

"The Arab world is different from ours," a spokesman for new French Prime Minister Jacques Chirac said this week. "Its reflexes of sentimentality and unity do not obey the same criteria as ours. The kind of operation that was directed against Libya risks provoking an upsurge of Arab opinion around him Qaddafi at the very moment when he is in difficulty."

With the exception of Britain's Prime Minister Margaret Thatcher, who was the lone supporter of the U.S. raid on Libya, most European leaders privately admitted that they bear a responsibility, too, by not having been able to unite behind an alternative policy against Libya and international terrorism that might have dissuaded Washington from going it alone.

One problem in finding a common European policy to confront Libyan terrorism has been, according to senior officials, Europe's basic revulsion to military solutions, a product of the traumas of World War I and World War II that by and large have left Western Europe convinced of the self-destructiveness of facile military solutions to very complex political problems.

On another level, there is a general belief in Western Europe, as there is not in Washington, that economic sanctions, as Craxi said today, are "totally useless."

Economic sanctions against Libya are also much easier for the United States to propose in view of its minimal economic interests there, than for European nations that have a much greater economic stake in the North African countries.

In 1985, for instance, Italy's trade with Libya was worth more than $4 billion, West Germany's was almost $3 billion and Spain's was $1.3 billion.

Beyond that, between 4,000 and 10,000 Italians work or live in Libya, although the number has dropped precipitously in the past month of crisis. About 5,000 British workers are still there, as well as close to 1,000 Americans and hundreds of other European nationals.

Italian private companies are also owed almost $1 billion in deferred payments for work done in Libya over the past years that any abrupt break in economic or commercial relations would endanger, causing up to 50 major Italian companies to face bankruptcy.

Despite those factors, most European governments insist that they believe Libya supports terrorism in Europe and elsewhere and that they would be prepared to back a policy that had a real possibility of solving the Libyan problem. They do not believe punitive attacks or efforts to assassinate Qaddafi will do the trick.

What might, say senior officials around Europe, would be an assault on the root causes of Middle East terrorism instead of just their most visible manifestations such as Qaddafi's support of antiwestern terrorists.

This would mean dealing with the Palestinian issue that, most European officials believe, the United States has sought to avoid either because it does not understand how much it is at the root of the terrorist problem in the Mediterranean or because of Washington's close political commitment to Israel.

The focus on Libya as the ultimate cause of instability and terrorism in the Middle East, according to Sandro Viola, an editorial writer for the Rome daily newspaper La Repubblica, has meant that the whole problem of the Middle East has been boiled down by the Reagan administration into the simple problem of what it sees as Qaddafi's instability.

But the real problem that is being avoided, Viola wrote this week, is the Middle East's general instability, expressed first in the Arab-Israeli war of 1973 and then in a wave of international terrorism that has its origin in "the Palestinian question which has been aggravated by Israel's inability to come up with a policy that leads toward a solution of the issue."