[Back after a busy week or two]
I'd say that both "If this Goes On..." and "The Handmaiden's Tale" are apocalyptic stories about takeover of the US government and society by extremist sects that are at least based on certain types of Fundamentalist Protestantism from the early 1900s Appalachian region and midwestern US. Heinlein's came first. The central characters and focus of the stories are quite different. Atwood's was on *women* and specifically one woman loosing her personhood, and then regaining it, despite her having forgotten much of her past and being enmeshed in the belief system at the beginning of the story. I find Offred, the Commander, *and* his wife all eminently believable and well-rounded characters. Her book ends with the main character's fate uncertain, throwing the book's focus squarely on the nightmare society it describes and not on what happens to that society.
Heinlein's book portrays a society that had lost its freedom due to fanatic fundamentalist religion, which he really loathed, *and* the successful revolt against that society and regaining of the lost freedom. It is IMHO not as written (heresy, I know) .
It's also a lot less depressing; I'm not sure Heinlein had it in him to write a genuine dystopia, as Atwood did. I don't identify John Lyle to the same extent that I did with Offred, but since I'm female and never went to a military academy, this isn't surprising. Lyle's character seems to me to betaken from central casting -- the bright, innocent, handsome young eagle scout manifesting in the middle of a dark and thoroughly corrupt military priesthood serving an even more corrupt and nasty prophet. (If you've kept up with the news about the Mormon polygamous offshoot group the FLDS, think of the priesthood and police department run by it in Colorado City, Arizona/Hildale, Utah.) The books spends more time talking about the overthrow of the system than in exploring how people lived, adapted, and *felt* while under it. I don't see either one as predictive of Donald Trump. by the way. Trump isn't either a true or false prophet. He is no sort of true believer in anything but Trump. He lacks the manipulation and acting skills to portray himself either as Nathaniel Scudder (a nasty but presumably believing prophet) or as the intelligent and corrupt Commander. Whatever you say about either Scudder or the Commander, they both showed *agency*. Trump like a lot of media stars reacts rather than acting; he's a projection of the desires of other people. Actress Lena Horne once said, about a small city that she found boring, "There's no *there* there." I've often thought that this comment applied better to certain people than to any city. It certainly makes me think of Donald Trump.