The Post delineates guidelines on this for consistency, largely, but also, I'd say, to reflect the broader mores of the public we serve. How such immigrants are referred to varies by media outlet, as Fusion's Felix Salmon outlines in a graphic at that site. Media companies clearly spend much more time thinking about this than the public at large -- but perhaps less time than elected officials.
How Democrats in particular refer to immigrants has changed over time. When the term "undocumented immigrants" came in popular usage, it was a replacement for "illegal immigrant" precisely because of the pejorative nature of "illegal" and the unfortunate habit of truncating "illegal immigrant" to "illegal." Over the past 18 years, the switch in usage by the party is obvious. (We used the Sunlight Foundation's excellent Capitol Words tool for the next two graphs.)
One of the earliest uses of "undocumented" was in 1996, by then-Rep. Bernie Sanders (D-Vt.), now, of course, a senator from that state. "There is a section of the bill," Sanders said that March, "that says that immigrants who are here illegally, or undocumented immigrants, are unable to receive health care through our public system." Since 1996, the term has been much more popular with Democrats than Republicans; members of the former party use it three times as often.
But Democrats use all three terms far less frequently than do Republicans, for whom immigration in general is a much more energizing subject. (That's why the graph above has a vertical axis that extends to 150 mentions per month: to allow for easy comparison with the other party.) Republicans are far more likely to use not only "illegal immigrants," but also "illegals" -- a term that has at times been used just as commonly.
There was a brief blip of the usage of "undocumented immigrants," but the expression has remained out of favor on the right.
It's hard to evaluate how Americans refer to the immigrant group simply because tracking language use in/by the public is a bit trickier. One way to get a sense for the terms people use is by seeing what they search for on Google. The graph below shows interest in each expression as a search term over time. And in that sense, congressional Republicans seem to be echoing the language more commonly used by voters. (This is an imperfect metric, we will readily admit.)
But again, most Americans don't spend a lot of time worrying about how they refer to people. The evolution of language in this regard is slow and marked by bursts of activity. (Think of the shift from "Negro" to "black," for example.) Democrats, responsible to a constituency, are headed in a different direction than Republicans, hoping that the voters will choose to join them on that path. Yes, I'm still referring to the language being used, but not entirely.