Spherical trig: huge one that was very much the norm, through WW2 (heavy navigation applications), now not a part of the standard course.
Solid geometry: still a part of the standard course, but emphasized more back in the day.
Projective geometry: (See bullet 2, but even more so.)
As for "real analysis", I think advanced analysis concepts (epsilon delta, continuity, etc.) were out there, if not emphasized the same way (and not a topological interpretation).
It's probably also useful to note that the Tripos emphasized a very computational, puzzle-y, problem-solving type of approach to math.
General editorial comments:
This is not all good or bad. It's probably still a mildly better course for people going into applied areas (physics/engineering) where "the 19th century math is alive and well" as opposed to the extreme rigor/proof emphasis that graduates students who can't solve AP Chemistry stoichiometry problems with two equations in two unknowns and a few conversion factors. Also, note that Ramanujan got massive inspiration from an encyclopedic mechanistic prep book for Tripos students (of the most abhorrent kind to those who obsess over abstraction at the expense of all else).
All in all, I think the way to look at "old school" stuff is as a source of hidden/forgotten gems. But not to rely on. Perhaps not to be returned to completely (for one thing the writing can be too flowery), and also of course, there are huge overlaps with current practice or things you miss, or don't want. But still dipping into the old stuff can be a great idea starter. Edwards Integral Calculus Treatise (a supposedly starter calculus course!) has freaking polylogs in it, with references to 1700s papers and such. And many problems can be found in old books that are good homework (for advanced students).
In general, older books (especially in English or from England) will click more with mathematical physicists or applied mathematicians and less with "don't want to do algebra because it's too trivial for me to do the work on" abstraction lovers and definition/caveat eagle-eye types. But even for them, the writing can be a hurdle if too flowery (a fraction are not flowery...but many are.)
Also, note that there are (usually) modern treatments of the less fashionable topics. So you are not forced to use books that are sometimes hard to read. All that said, I think occasionally dipping into old stuff is a good way to find ideas.
P.s. I think it's very fair to find many areas where the course has not changed much. If you had a great knowledge of algebra 1, 2, trig, geometry, and first year calculus, and ODEs/PDEs, you would be just as good as someone from nowadays at least in terms of engineering math. Maybe within that course, the one thing lacking would be strength in vectors (not even vector analysis) as it is emphasized more now and earlier.