Differentiated gameplay across classes would require a few things, IMO.
For one thing, no shared spellcasting lists. Imagine a game where Necromancers cast necromancy spells, that's it. Illusionists only created illusions. Maybe a generalist wizard in there, but casts much less efficiently due to spreading her focus.
Different spellcasting methods. Anyone else remember the Channeler, Defiler, and other options from the 2e S&P and S&M books? Something like that, but really differentiate them. Maybe Channelers lose HP to cast (there's a wizard in the Sentinels of the Multiverse game that works like that), maybe Defilers damage the area and characters around them to cast spells. Might create some neat in-game decisions...
(As an aside, I know this isn't about caster supremacy, but early on the convo was about how reliable spells are; I agree, to the point where they feel more like science than magic. Robust systems like channeling and such would make them feel special again, IMO. If it's about supremacy in combat, what about re-introducing spell disruption?)
A set of maneuvers that for martial classes resemble the philosophy of video game mechanics. In "Injustice" my son rarely beats me, BUT...if he uses Martian Manhunter and manages to pull off one particularly difficult move, he can chain it to a bunch of follow-up moves that decimate me.
Imagine each martial getting a set of maneuvers like that, wherein if you pull off A (could be gated by a particular d20 roll), your next attack can be B (which gives bonus Z, a condition maybe), and if you get that, you can do C (which gives bonus Y, a heavier debuff?). Seems to me there could be a lot of unique designs in here, tailored to each class, making each one feel unique.
Spells that augment skills rather than replace them entirely. PF2 seems to have accomplished this, according to Campbell.
Just my first thoughts on the OP topic...there's some potentially really interesting ideas to be mined from this topic.