Outside of a few complex SPAs doing interesting things, structurally we're building the same things we were building back then (blogs, brochure sites etc) — it's just that the way we're building them has become (optionally) much more complex.
IE 5 and 6 and Netscape Navigator 4 would all beg to differ. Having made simple websites two decades apart, CSS is much less hassle nowadays.
I know enough of Web FE development to jump in and fix a couple of bugs when needed, and on my own consulting gigs as side job, I only do native desktop development as kind of therapy.
Personally I think of the web as a highway: great place for a billboard, not such a great place to be storing or mutating sensitive information.
Most websites could be a simple WordPress do. Many brochure websites should be static. There are benefits to straying from that, but they come at a cost.