The main difference is in their range. Both specialize in charged attacks; whilst Yanfei drops an anvil on top of the enemy from a pretty decent range, Klee fires a projectile at a downward angle and can't seem to hit very far (from what I've heard). Even with normal attacks, Yanfei throws a targeted fireball with significant range, whereas Klee throws an arcing projectile, and thus has more limited range (and also it deals like 3 damage apparently).
EDIT Thanks to SPArcheon for reminding me of this as well. Klee also has a ton of animation cancels with just about anything she does - her slow animations actually caused her to be considered bad for some time until people learned the cancels. On the other hand, Yanfei doesn't really have that. Her animations are a bit faster, but you can't cancel as nicely. Her normal attack comes out quickly enough that there's no point cancelling. If you jump or sprint after summoning her anvil (charged attack), it will just disappear (there's actually even a special animation for the anvil dissipating from being canceled). You can still cancel it if you wait until just as the anvil starts falling, but her charged attacks are fast enough even back-to-back that there's no point.
Yanfei also has ~~Star Jades~~ Scarlet Seals, which is a mechanism Klee doesn't have. Doesn't really change the similarity in their playstyle but just something else to note.
Klee's charged attacks are bombs whereas Yanfei's is just Pyro damage - the distinction is that Klee can break Geo constructs and shields (thanks to Steven for this point). I forget about Klee's normal attacks and whether or not they deal AOE / are bombs, but Yanfei's normal attacks are targeted and only deal Pyro.
Klee's E is a deployable, whereas Yanfei's E is a burst ability.
Their bursts slightly differ in that Yanfei buffs herself but Klee just deploys damage automatically; this is not too significant of a difference since both just increase damage and force you to keep them active lest you dispel your ultimate early.
Their ascension passives aren't different by a significant amount - Yanfei gains PYRO% and deals bonus damage; Klee deals bonus damage and reduces stamina cost, and gains Energy for her party.
Finally, their constellations pretty much just do damage, more damage, and even more damage.
So, overall, I agree with you that there's very limited differences. Their ascensions and constellations are basically just "damage and a ton more damage", and their abilities differ pretty much only on technicalities.
TL;DR: The main obvious difference that I noticed right away, even only having played Klee during story quests and character trials, is that Yanfei has significantly longer range, Klee requires more timing with animations whereas Yanfei just has smoother animations, and Klee can break Geo constructs and stagger a lot more heavily than Yanfei can.