I wonder whether there exists any difference between humblebrag and false modesty. E.g., is false modesty more general than humblebrag, or are the two expressions synonymous?
Firstly, false modesty is an uncountable noun, and humblebrag can be either a count noun or a verb.
The two terms are certainly related, but to me, as a native US English speaker, there are some subtle differences. False modesty is, quite simply, being modest or downplaying your accomplishments in a way that is not true modesty, maybe to call attention to them, or to encourage people to keep praising you.
A humblebrag is a more specific kind of behavior about calling attention to yourself by saying something that looks modest or self-deprecating (the "humble" part) but contains a self-praising subtext that is the real message (the "brag" part). For example,
I hate my lambo! Police is ALWAYS pulling me over just cuz its a lambo so they always think I’m speeding but I’m not!! Then they let me go!
The surface message is "I get pulled over for speeding and I hate it", but the real message is "I have a Lamborghini."
A humble brag is when you say something that sounds like it is a bad thing but is in fact a good thing for example
Oh my dress, it is quite an old <really expensive designer brand) dress
so they start off saying they are wearing an old dress but end up actually pointing out they are wearing a really expensive designer brand dress.
false modesty is much wider and can refer to anything, like deprecating your own achievements, or saying you couldn't do something. It isn't you pointing out something good, which is why it isn't the same as a humblebrag.
person 1: could you update the widget on the thingumy
person 2: oh no, I wouldn't know how to do that it is far too complicated
person 1, but you designed both the widget and the thingumy didn't you?
noun [ U ] UK /ˌfɒls ˈmɒd.ɪ.sti/ US /ˌfɑːls ˈmɑː.dɪ.sti/ behaviour in which a person pretends to have a low opinion of their own abilities or achievements: He shows great pride in his work and has no false modesty about his success.
First, about modesty and bragging.
Modesty and bragging are opposites.
Modesty is downplaying your own achievements or abilities, almost trying to make them go unnoticed; because you think others' are more important, or just don't like being the center of (undeserved, in your opinion) attention.
Bragging is boasting your own achievements or abilities, making sure they do get noticed; because you want to feel superior to others, or just like being the center of attention.
Now, to your question.
False modesty, just like false currency, looks like real modesty: it is still downplaying things. But, just like false currency, it's origin is not "true": false modesty happens when someone fakes modesty to be polite, but they still think greatly of their achievements and like being the center of attention.
False modesty is not necessarily bad -- as I said, sometimes it's just a way of being polite. However, it can be seen as something bad if it is "too fake".
Humblebragging is still a kind of bragging. You're still calling for attention over some fact, achievement or ability that you want others to notice. However, you're doing so in a covert way, disguising it in a more "humble" scenario; by doing so, if someone accuses you of bragging, you can always say that you were not actually bragging: you were just talking about the humble scenario, it's the other people the ones fixating on that thing they are accusing you of bragging about.
Humblebragging is usually regarded (IMHO) as a worse trait than false modesty, because there's no politeness in it; it's just bragging with an alibi.
In that sense, "false modesty" is still (faked) modesty, and humblebragging is still (covert) bragging.
Since modesty and bragging are opposites, false modesty and humblebragging cannot be synonymous.
Let me just add some historical context, if that helps at all. The phrase "false modesty" is at least 150 years old. (I don't have a specific origin date, but several famous authors used it in the mid-1800's, if not before.) Conversely, the term "humblebrag" has only been around for just over a decade; it's creation is attributed to Harris Wittels (executive producer of the NBC show Parks and Recreation), on Twitter in November 2010, where he apparently gave several examples. It was added to the "Urban Dictionary" four months later.
With the advent of social media, there are of course many opportunities to do either. How many IG posts are there where the young lady complains "how bad her hair/makeup/clothes/body look today," complete with photo spread, only for the purpose of reaping compliments? In this sense, I don't think you have to be anyone special to have false modesty.
However in one of the examples given by Wittles, "I just realized I have only showered in one of my five showers", that does seem to require that you actually do have something to brag about, even if you are doing it "humbly".
false modesty (see "References in classic literature")