Ok, so I did some research on sekai-kei as a genre.
The TV Tropes summary: http://tvtropes.org/pmwiki/pmwiki.php/Main/SekaikeiGenre?from=Main.Sekaikei
An often-cited scholarly article speculating on why sekai-kei appeals to the psyche of modern Japanese young men: http://www.e-ir.info/2013/02/01/apocalyptic-imagination-sekaikei-fiction-in-contemporary-japan/
It's not terribly long.
(As a side note, I have a fundamental disagreement with the argument that Japanese young people are withdrawing and deviating from the norms of adult society because they aren't "emotionally mature" enough to commit; there are most likely economic and social factors as well. In America people criticize millennials for not settling down with a marriage and a house in their 20's, but that's because student loans, job scarcity, and the inflated housing market make it literally impossible for most of us to do so, and I have a feeling we're not the only ones on the globe to have this problem. Any lingering immaturity could be correlated but that does not necessarily imply causation. /qualifying anthropological rant)
If we assume that Neon Genesis Evangelion was the work that launched sekai-kei, though the genre seems to have transformed over time, that gives us a distinct emotional atmosphere. There is a tangible existentialism and at times a genuine hopelessness for the future, with flawed characters and an open ending that you can't really call "happy." The article maintains that another important facet of sekai-kei is minimizing the "community" connections of the wider world to focus on two extremes, global crisis and direct interpersonal relationships (specifically romantic ones). In Shin Godzilla, not only are the characters all adults with adult responsibilities, there is also a focus on the relationship between politicians and the wider diplomatic community, and the feeling that actions will have lasting international repercussions, which is apparently another deviation from the genre. Though the movie does have the hallmark mysterious unexplained threat and focus on individuals' emotional reactions to crisis, it is definitely unconventional, especially considering both were directed by Hideaki Anno.
Another work commonly defined as sekai-kei is Voices of a Distant Star (Hoshi no Koe), another Makoto Shinkai movie. If we focus on the differences between these two movies, there is a definite change in tone. Hoshi no Koe is brutal in its depiction of isolation, with literal years passing between communications. If I remember correctly, the main couple are the only characters that appear in any detail, possibly due to budget constraints as much as stylistic choice, but it leaves us with a barren emotional landscape. It's a surprising subversion to have the girl going off to war and the boy staying home to wait, but with the focus on him it becomes a lonely personal struggle with waning faith rather than an intense space battle. The ending is open enough to be contentious, and if you say you didn't cry you're probably lying. Comparatively, Kimi no Na Wa is grounded in its community, location, and heritage, and has a much more uplifting kind of message. There is still loneliness, the idea of meaningless random encounters easily forgotten and the possibility of an indifferent universe, but there is also the implication that might be some kind of order or meaning to the world, or that we can create one. I can really only describe it as a hopeful movie, which is probably part of what made it appealing to a more mainstream audience, as opposed to Shinkai's earlier works.
It's kind of fascinating to think about both director's earlier vs. later works in this way, but I don't necessarily think it implies "personal growth" on their parts so much as maybe a desire to tell a different kind of story. It seems like the biggest differences are a more hopeful view of humanity and the universe and the inclusion of more (and more varied) characters vs. focusing on one or two characters' emotional journey. I don't really see one as superior to the other (it doesn't really feel like "selling out" to me) but my brain is fried at this point and I'd be interested to hear what others have to say. That is, if you can get through my tl;dr wall of text lol