long story short, original staff wanted to end it with the first movie, Nick said no, so original staff quit one by one. Nick then had the brilliant idea to replace the old staff with a chunk of the OG Ren & Stimpy crew, which is probably where things went "horribly" wrong. This choice is directly responsible for the toenail scene, that cat, and everything else /co/ likes to bring up in threads like these.
What isn't brought up is this: Immediately after the movie (when Hillenberg stepped down as showrunner and many of the OG staff left), the sound editing department got restaffed (and likely defunded.)
This is the number one reason the show lost its touch from season 4 onward--even before the out-of-place R&S humor and the mediocre writing, the sound team from the very first post-movie episode onward completely changed the tone of the show, very much for the worse.
The reason everything feels strange in Fear of a Krabby Patty, despite it being a solid enough episode (C.H.Greenblatt stayed on till season 5-ish, then left to make Chowder) is because of the sound.* The trademark comedic timing is strangely lacking. Lines feel too close together, like there isn't enough time between them for the characters to process what is being said. Later, music starts being used as punctuation instead of as background, and pacing issues make this all the more obvious and jarring, further compounding the feeling that something is just WRONG. Inter-movie Spongebob has tons of problems, but the defining quality is the shitty sound. It's how you can tell a new episode and an old one apart.
On the bright side, since the second movie, the R&S crew is moving on and the original staff has started to return, so we might get *somewhat* back on track. The audio still sucks, though.
*yeah, the animation is hella off model too, but that doesn't kill an episode by itself. The Pretty Patties episode is seriously off-model, and it's fine.
(addendum in next post)