I'm very leery of (collaborative) open-source software which is supposedly better than much more polished commercial software full of "bloat and eyecandy". For a long time I avoided LaTeX, since I assumed the users were just being elitist about an overcomplicated system (needless to say, I no longer hold this opinion). Even so, I have recently (after the release of Word 2013!) started using LaTeX where possible. This was explicitly due to killer features LaTeX has, that Word not only doesn't have, but likely cannot possibly have.
I still use Word for quick, short, simple documents. The extra overhead for writing and maintaining documents in LaTeX unfortunately makes itself felt. But I've learned the hard way that there is a threshold (usually around 10 pages for me) after which it's best to give up on Word and switch over to LaTeX.
Word is capable of automatic equation numbering, index generation, table of contents and so on
Yes, but only insofar as it allows you to claim that it does in this exact sort of discussion, and compels me to begrudgingly accept that yes, technically it does.
However, these features are extremely basic. If you're a caveman who has just heard of ToCs and want to include one in your new great Pangean novel, it will work. But realistically, many texts are written to strict requirements. There are many nuances of formatting, style and just the particulars of how these are generated that Word is unable to do. There are many more that Word is able to do, but it is so laborious as to be impractical.
Point and click is nice for using features for the first time. But once you label your 1000th figure, it becomes very annoying that you can't just press some simple key combination to quickly get a label that looks exactly like you want (unless you happen to want the default one).
In Word you can easily define templates and structure your text using defined levels of headings, etc.
This has, in fact, been possible at least as far back as Word 2003. The problem with styles in WYSIWYG is that there is no longer a 1-to-1 map between appearance and style, so you can't always easily tell exactly which characters are using which style. For instance, is this period italic (it's not irrelevant - if you start inserting characters after an italic period, Word will make them italic)? Is this space bold? Is this bit of italic text actually using the Emphasis style, or is it just italic (Word sometimes can match the styles automatically, but this is not reliable)?
On the other hand, when editing markup there is in fact a 1-to-1 correspondence between what you see on the screen and the actual structure of the text - because every formatting command will be appropriately indicated in the source (which is what you will be working with anyway). The trade-off is that you can no longer easily see the appearance of the output easily (exception: features like the buggy in-line preview in TexStudio), but it turns out that for many documents, it's not hard to "imagine" how a given bit of TeX looks after you've been working with that document for an extensive time.
Also, a word on defaults: The default styles of Word tend to be inadequate. To my untrained eyes, they are only mediocre. My trained graphic designer friends say they are hideous. The extra styles you could look for on the internet and download are, for the most part, also not very pretty. Conversely, even a default document in TeX looks gorgeous. Sure, you could easily make a "LaTeX style" in Word 2013 - but as a user who hadn't seen how nice LaTeX looks and isn't trained I had assumed that professional typesetting can only be done by large publishing companies, and never even tried with Word.
Word 2013 is much more stable for large documents than it was 10 years ago
Indeed, in my experience it hardly ever crashes. In that sense it is stable.
Performance is a whole other matter. With TexStudio, I see hardly any impact of length on performance. Even compile times can be kept manageable thanks to
include. With Word, after adding a couple dozen figures and tables, a few hundred references, typing into the middle of the text leads to very perceptible and distracting lag.
Word now has an improved equation editor with nearly plain-text Unicode editing and customizable keyboard shortcuts.
The equation editor in Word is unfortunately crap. Again, typesetting a very basic equation for the sake of being able to say that yes, Word can technically typeset equations is possible and easy.
But once you start trying to do real work, you quickly notice gaps in functionality, both in terms of what forms can be displayed and whether the tools provided are effective at saving you time and effort. Add to this the issue I mentioned above about seeing exactly what the input markup was from looking at the output, and it becomes challenging and frustrating to get your equations looking just right. Sometimes (like a simple sheet of notes from a non-math talk) it doesn't matter if the equations look perfect. But other times (like writing a paper) it may be very important or even mandatory.
As for keyboard shortcuts - well named TeX commands are mnemonics of themselves, and if your editor supports auto-completion, things become much easier. Keyboard shortcuts have to be laboriously memorized and are less straightforward.
What are major improvements of LaTeX or associated tools (e.g., editors) which fix some (major) disadvantages LaTeX might have had in comparison to Word some years ago? or introduce new (killer-) features where Word can't keep up?
I think development of LaTeX editors like TexStudio has made working with LaTeX much easier for many users, as well as drastically lowering the barrier to entry. Sites like this one have made getting help much easier.
One killer feature that was a big factor for me, which you don't mention, is working with parts of the document at a time. If you have a long book, and want to work on one chapter at a given time, Word does not make this easy. Keeping the whole thing in one file causes lag and reduces the usefulness of the scrollbar. You could have separate documents, but then how will the page numbering of
Book - Chapter 6 know to stay in sync with
Book - Chapter 5? How will you keep the styles consistent? If you edit your document a lot, and accurate formatting is critical (e.g. grades taken off for formatting errors), just validating the formatting of your document becomes a headache in it's own right.
A second one is figures. The one Word feature I miss in TeX is being able to drag and drop JPEGs to exactly where I want them - but aside from that, Word is very annoying in how it jiggles your figures around the text when you add content in preceding sections, or how there is not a tight coupling between figure and caption (which is still just an ordinary textbox, that the caption tool positions for you initially).
Another small one is citation styles. Word has a few citation styles included by default, but there are glaring omissions (for example, the style used by Nature). You can luckily download some common ones if you search online, but if you still cannot find the one you need, making your own citation style is not a simple matter.
Speaking of citations, the built in reference manager in Word is horrendous for anything beyond a handful of cites. If you have more than 5 sources, the busywork of filling out a whole form for every source really adds up, meanwhile tools like JabRef can just fetch all the information by parsing output from sites like Google Scholar. I'm not sure if this counts as a missing feature, since it is solved by those same TeX tools - for instance, JabRef can export citations for Word, effectively nullifying this problem.