Chidi's sideburns are slightly longer after he and Eleanor step through the green door and appear in Athens.
Item 9359 of Tahani's To Do list is "Solve the Poincaré Conjecture". Apart from a minor point that it's a proof rather than a solution which is required, the conjecture was finally proven by the Russian mathematician Grigori Perelman in 2006.
Earlier in the season Tahani said that she had another godfather compared to the guitar player mentioned here. However, Tahani had already mentioned multiple other godfathers in other episodes, so regardless of the why that's the case, it clearly wasn't an accident that she mentioned more than 1 additional godfather in this episode.
In the opening scene, Michael is recording his song on the guitar, but the recorder is neither working or set up properly. The VU (Volume Input) is not registering any input, and the top right meter's input is stuck at max. In addition, the tape used to record the audio is not connected between the two reels, meaning there's nothing being recorded. You can see the end of a piece of tape flapping on the left reel as it spins.
When Michael is playing his "Marshmallow Cat" song, he tells Janet that he can't get the "E" chord, yet the song ends with an "E" chord which he plays flawlessly.