The difference is that immigrants to the US melted their culture in with the existing one, creating the America we have today. Everyone adapted. People came to the USA to become...Americans. One can quibble about hyphens but at the end of the day it's always "-American".

I think this is a very ahistorical misrepresentation of what happened with successive waves of immigration to the U.S. For example, the Germans - who you cited above - very much did not "melt their culture with the existing one" when they arrived. The really big waves of German immigration lasted just under a century, starting in the 1820's and peaking in the 1880's, and they were for decades the largest group of non-English speaking residents in the U.S. and accounting for nearly 10% of the total population by 1900.

And they weren't just becoming....Americans. They lived in "Kleinedeutschlands" all across the U.S., almost entirely retained their own language, had German-language schools and newspapers and other community institutions that they maintained to a great degree. They changed American culture as well, which is why we have things like Christmas trees and kindergartens and things can be as American as "hot dogs." Etc.

It wasn't until the Great War, when American started hating and persecuting German people, that this changed. Only when people started to discriminate against Germans just for being who they were did they feel that they had to hide that, and start abandoning German language and cultural markers. Not because they were any innately prone to "melting" into the culture than modern immigrants (they weren't), or because they wanted more to give up their heritage (they didn't) - because they were being persecuted and had the ability to hide because they weren't ethnically distinct from "real" Americans.

The same is true of a lot of the other immigrant groups. No one at the time thought that groups like the Irish or the Italians had the same culture or values as good ol' fashioned 'Mericans. They were also groups that were foreign, alien, had cultures and practices that were too different from true American culture to be accepted into our society - and which formed their own neighborhoods and societies and institutions and schools and were isolated from mainstream America until persecution and lack of access to economic participation forced many of them to give up (under that duress) their culture.

Here's a neat trick for looking at it: for everything you see in Western Europe having issues with Muslims, the historical equivalent here is Americans having issues with Catholics - an alien and foreign religious group with such divergent values and cultural practices (to say nothing of linguistic differences) that they will cause nothing but trouble if we let them in. Funny how history rhymes, and love the irony of a Buchanan or a Tancredo leading the charge against the immigrant problem...