Generally speaking, as far as we can tell anyway, the public were supportive of the idea of Japanese -American internment. There had been a long history of prejudice and suspicious towards Japanese immigrants - California in particular was notorious for this fact, and had numerous anti-Japanese organisations, such as the 'Japanese Exclusion League'. They successfully campaigned for the signing of the Alien Land Law which targeted any Japanese-American living in California who wanted to own land. Historians such as Alice Yang Murray and Roger Daniels have argued in their own works that the Japanese Internment was simply a culmination in the decades long prejudice that was "fuelled by economic competition and racial stereotypes".

During the period of internment, flags and posters proclaiming areas as "whites only'' and telling "Japs to keep out" were all too common during this period. You didn't really see any public backlash from White Americans, as they saw Japanese immigrants as the traitors who had brought about the attack on Pearl Harbor, and numerous community leaders who had been placed under surveillance were accused of giving information to the Japanese army.

At higher government levels, there were some officials, such as Attorney General Francis Biddle, who privately disagreed with the Executive Order as they knew that there was no concrete evidence to suggest that any Japanese-American citizen had given information that aided the attack on Pearl Harbor. However, they did not publicly voice their concerns when it became clear that Franklin Roosevelt was in support of the idea of Japanese internment. These were in a minority however, and one extremely vocal supporter of internment, General John DeWitt, made frequent statements on how "a Jap was a Jap, it makes no difference if he is an American citizen" and proclaiming them to be a traitorous race. What is interesting to note about DeWitt in regards to internment is the fact that he wanted to include German and Italian-Americans, but was over-ruled; no doubt due to the sheer number of descendants that both nationalities had across the United States. EDIT TO ADD: yes German Americans were arrested and held but the difference in numbers is vast, over 110,000 Japanese Americans were interned for the duration of the war, whereas it was around 11,000 German Americans that were detained.

Japanese-Americans did obviously oppose the policies however, and there were cases of them resisting the Executive Order. In Korematsu V. United States, Fred Korematsu (who was later awarded a medal by Bill Clinton), went to the Supreme Court. However, the Court ruled in favour of the government and officially gave a legal precedent to the Executive Order. There were hundreds of people who resisted and had to be forcibly arrested and relocated against their will, however there were also many who believed that it was their patriotic duty to remain in the camps for the duration of the War - they were told by the War Relocation Authority that this was their contribution to the war effort. Executive Order 9066 remained a popular decision in the years after the war and Franklin Roosevelt remains an incredibly popular president. It took until the 1960s/1970s for the government to admit that the internment was wrong.

I've noticed that the interment of Japanese in America is pretty much the only kind talked about. Was the Japanese experience very different to that of other enemy aliens, or is it more of a historical memory thing (I also know very little of the details of what happened)?

Originally, in December 1941, the public was supportive of the Japanese, especially of citizens. This lasted for about a month or two when public opinion slowly turned. In particular, the Robertson Report came out and the Niihau Incident was investigated and publicized. The former found evidence of intelligence gathering aided by the local Japanese populations. The latter was an incident in which two Japanese Americans attacked other Americans to aid the escape of a downed Imperial Japanese pilot. Absent racism, these incidents would have justly led to treason charges against those directly involved. It might have justified a program similar to German and Italian internment, where specific individuals were brought before judicial hearings and interned with due process.

Instead, partly due to panic and partly due to racism, support for general internment grew. The majority of the population rejected explicitly racist arguments (though they got some voice in the press). Still, a small but vocal minority said them openly and there's little other explanation for the disparate treatment. Likewise, the Japanese-Americans themselves reported increasing amounts of discrimination and racist attacks. Some newspapers even published guides on how to tell the Japanese from other Asians so people could direct their racism properly.

The United States established military districts over much of the West Coast. All those descended from populations of the Japanese Empire were required to leave these districts or submit to deportation. Notably, this included Koreans, Taiwanese, and other imperial possession. These western states were the only place where internment occurred and, in fact, the majority of Japanese, Koreans, etc. were never eligible for internment. Tens of thousands of Japanese chose to leave the zone instead of submitting to internment. Sometimes the government offered to resettle them outside the district altogether. Many students escaped by transferring to eastern colleges, as several colleges had sympathetic boards who encouraged this process. Those who remained behind were moved to camps.

There was significant opposition even at the time. Several protests were held and some states introduced legislation to prevent deportation. Only one, Hawaii, actually passed it (and in doing so saved more Japanese from deportation than were ever actually deported). There were several arbitrary arrests which were often condemned in newspapers, even as public support grew. In the end, though, this was a vocal minority and deportation eventually proceeded.

Still, it would be fair to say the action was controversial. The United States felt the need to justify and explain it, often explicitly responding to those who claimed it was illegal or anti-democratic. Propaganda films were produced and various documents published. By and by large, the tone is that the majority of the Japanese are loyal citizens who are sacrificing for the country, just like the rest of us. Racism was, several times, explicitly disavowed. This does, unfortunately, not make it so. And several people high up, like Attorney General Biddle, privately expressed that they thought internment had racist motives. The heads of the camps themselves quickly grew critical of the program and several administrators transferred out or produced memos condemning either specific rules or conditions or the entire program.

Still, public support ran high in 1942. As fear and fervor died down, so did support. By 1944 it had waned and people were publicly criticizing the camps, even within the government. The government itself was increasingly on the defensive to explain its actions. And in December of 1944, the Supreme Court handed down the unanimous Ex Parte Endo decision, declaring the camps illegal and a violation of civil rights. The government was permitted to intern people only upon proving disloyalty, as was the case with the Germans and Italians. The program, by that point, was unpopular and even the administrators of the program were eager to assist in its end. Despite this, 1945 was an election year and the government chose to set a date after the election to avoid political fallout. (This was, notably, objected to by the administration of the camps who wanted them closed as soon as possible.)

Those interned were first permitted to return home in January 1945. Some stayed as late as 1946, with the government having to push stragglers out. Many returned home to face discrimination and the government offered only some, largely insufficient protection. Notably, the government also offered compensation but required that the interned provide evidence. After a rushed evacuation and the loss of IRS documents, few could prove serious financial loss. Those who did were duly compensated, but it was a small number. What few defenders the program had after 1945 dropped off even further, excluding those who saw it in exclusively racial terms.