Thinking it through, I've decided that Heinlein and Meyer are largely similar as writers, in terms of underlying approach and why they have their popularity. This is a somewhat counter-intuitive judgement. After all, Heinlein was a foundational writer in 'Golden Age' science fiction, noted for contributing to the focus on sex and violence, and generally resistant to sentimentality. Myer writes young adult vampire fiction with a strong romantic focus. The way they're marketed is different, the direct subject matter is significantly different, and there doesn't appear much overlap between the two groups of fans. Most Heinlein enthusiasts I've seen snarl at the mentioned of Twilight, and I suspect the typical group of Twilight fans would dislike Heinlein's classics. What is the commonality then? Well, they're both popular authors that I find grotesquely overrated, but that's not sufficient grounds. I dislike Twilight and Hylozoic, but see very little similarity between the two. Heinlein and Meyer are ultimately more complementary.
It's somewhat revealing that the similarity is rejected so forcefully, I suspect most Heinlein fans would snarl at the prospect. Certainly it's not that Rico and Bella are identical in the main positions of their life, the external focus are very different. What's similar is the lack of a real personality, the way they are in fairly character-oriented books but are without much defining personal detail, larger interests or a sense of inhabiting a world beyond the pages. The main advantage to this is it enables the reader to construct their own position on the blank slate that is the protagonist. Like in some Role Playing Games, one is given a name, a set of skills and a larger story to follow through on, but the more direct aspects of who the person is remains undefined. The narratives are largely open-ended, ambiguous in a key presentation even while they're very insistent on key arguments elsewhere. Obviously there are major differences in the stories, genres and way the two are constructed, but someone that rails against Meyer's poor characterization while taking Heinlein as some kind of model for effective characterization is being inconsistent. This doesn't necessarily say that the two writers are on the same level of quality, and as written works Heinlein's are aesthetically better. Both authors are monumentally flawed on aesthetic grounds, and on the level of compelling well thought out ideas, which suggests there are other components behind the popularity.
The structure behind the two authors writing for this is obviously fairly different, but also complementary. I take both Heinlein and Meyer to be misogynist writers who presume and validate the primacy of men in their idealized narratives. Male drive is taken as the force behind events, masculinity is the basis of public action and power, and unstable male behavior is accepted as an acceptable price for adoration. Where the two authors differ markedly is what they take as the focus of their stories from that point. Meyer is all about femininity within this male-dominated structure that her stories accept: the female protagonist is passive, self-injuring, self-loathing, alienated from her body and in continual need of rescue. The rescuing male is highly creepy and intrusive, but that's accepted as a sign of love. Heinlein takes the standpoint from the perspective of men, showing the protagonist as rough, super-competent, generally a sex magnet, generally witty, whose success with women is justified by the sheer possession of machismo. Heinlein did a lot to introduce sexuality into science fiction narratives, and it increasingly became an obsession in his late writings, particularly through the form of incest. I'd say that for Heinlein particularly late in his career it's not about casual and nonconventional sex as leading to any type of societal shift, challenging norms in general. Sex is a matter for individual men to exercise and achieve, with the behavior of the masses not really a matter of concern for his stories. Nor does it even really seem fuelled by hedonism, in the way it's generally understood. I'd say it's about a man being able to get a woman any time he wants, a variety of women, up to and including blood relations. That marks him as the ultimate model of virility, power, superiority. It's tempting to read Heinlein's stories as directly sexualized and polygamous while Meyer's are non-physical and monogamous, but that contrast doesn't really hold with Heinlein. For all his exploring themes of group marriage and group sex, in an equal number of his stories he ends with pairing off a couple and settling things down rather conventionally.
In a strange way this brings it back to Twilight. That book took heterosexual monogamous love to blatantly idolatrous extremes, famously teaching that a woman should cherish a man's love even when it was accompanied by stalking and menace. Heinlein doesn't prioritize romance, or even sex, to anything like the same extent, but when he does it's not exactly free of dysfunction. His books teach that a woman can still have a stable and happy monogamous relation with a man, even if he belittles her, tries to kill her or rapes her. Ultimately Meyer and Heinlein's gender representations aren't as far off from each other as they may seem, and can actually be quite compatible. That's perhaps the best indication of how screwed up both standards truly are.
Mostly the discourse on Heinlein as against Meyer seems to emphasize the skill of the former more--since he was such an influential figure in the genre, since he ostensibly had such interesting thinking on politics and political systems. Meyer is seen as just a YA vampire romance writer, and particularly as a lot of the mass cultural critique of her turns on her belief system she's assumed to be more naive and thoughless in her writing. Looking at the two I find it just the opposite. Heinlein plainly puts himself on the page a lot more, and a lot of his main world systems are the expression of different contradictions that he seemed to be trying to work out. Meyer's main story seems the more controlled, the more calculated, the one focused on delivering a specific type of formula in a new and stronger fashion and appealing to a specific subset in the population.
One could say that Heinlein is the utopian spirit within capitalism, producing the narratives that sell an image of heroic individuals fighting for their right to market forces. Meyer's Twilight, in contrast, in the working of the capitalist process on the level of genre fiction more consciously, and without illusions as to what it's doing. As a book Twilight aims to produce a certain fantasy that will appeal, and provides as much plot and as little character as appeals to that end. In it's content it's precisely about living in unrewarding economic situations without the dream of libertarian betterment and heroic uber-masculine success. Instead it pivots on a fantasy of rescue through the technology of romantic infatuation. In a world coded for despair and psychological under-fulfillment, utopia comes from outside the self, outside the conventional economy, outside the natural. The Cullens are wealthy, aristocratic, but beyond that they exist outside the regular economy, offering a non-religious salvation that transcends physical boundaries. In that sense it's not surprising that Heinlein can't get enough of politics while Twilight is entirely void of it.
The anti-Twilight standpoint is interesting to look at in this light. There are certainly things worth criticizing in it, but I think a lot of that standpoint has been already outmaneuvered. There's no magic bullet of criticism that will convince a fan that the work is fundamentally wrong and twisted. There usuaully isn't, that's not how reviews usually impact, but it's particularly difficult in this case. Share with a Twilight fan the idea that Edward is a sick, dangerous puppy that should be avoided and that Bella is suicidally reckless for trusting him? That belief is already in the text, expressed within the narrative. It's quite extraordinary. If these viewpoints had showed up in later volumes, after Twilight became a huge phenomenon. It would seem then an obvious after the fact way to block against the dominant criticisms. But those are built into the template from the beginning, as if Meyer anticipated the amount of success it would gain, and the backlash, and so she went out of her way to build in the anti-Twilight story within Twilight.
There are a number of implications of this, perhaps the most pressing is suggesting a new approach to framing the anti-Twilight discourse. Much of the structure of this rhetoric uses transparently weak grounds: focusing on the Twilight readers as ignorant squeeing fans incapable of proper thought, or focusing a xenophobic distrust on Meyer's status as a Mormon. In alternative, it may be useful to understand Twilight not just as an uncountably popular failed book with a dysfunctional gender message, but as a well-calculated and successful book that is working to produce and replicate certain specific things. Analyzing the rebuking the flaws of Robert Heinlein, in aesthetic skill as well as representations of men and women, may be a productive approach to a fuller gender analysis, and to framing a more effective response to the appalling misogyny embedded in Twilight.