Firstly, I think that you made an error in the first sentence. It should be: I am under no illusion that living in America is going to be very easy. You should avoid having a double negative (in this case "no" and "not") in a sentence because then the meaning becomes vague and/or confusing.
The two sentences have a similar meaning with this correction.
You would use the expression "I am under no illusion that" when somebody suggests that you believe something that is not true and you correct them by saying that you don't believe that something.