The problem with the word "integrate" is that it puts a lot of the responsibility onto the home community. Yes, they can integrate into the community, but it's also on us to integrate them... meaning we might have to accept all their customs, even the things kesstrel mentioned. It implies they should be able to live here almost like they lived at home.
The word "assimilate" infers that they should take more of our values instead. And this really is needed to avoid a fragmented society that can give rise to fundamentalism.
I hate the fact, for example, that Italy had to cover up nude statues (?!?) for the Iranian president's visit. I hate the fact that he wouldn't even dine with his French counterpart if there was wine on the table. I mean... WTF... you're visiting a country, you can't expect that country to bend to your wishes!!!
See, one of my closest friends is Muslim. She drinks wine. She lets me eat pork if I want to. She sends me Christmas cards. She definitely doesn't expect nude statues to be covered. And yet, her family is religious. They are people who live by our norms but haven't forgotten their roots. However, they are from the Commonwealth... and I guess that helps a lot.