The main focus seems to be on Twilight realising everyone has a talent and unique place. However, we also learn that within all of Twilight's angst of her screwing up the ceremony, every single other pony has been doing just as badly as her, and the supposedly blissful Winter Wrap Up has been a disorganized mess before she even came. As such a second Aesop could be "Everyone makes mistakes, so don't feel singled out because you made one".
Alternatively: Managers and organizers are the ones who were Kicked Upstairs because they can't do any of the real work.
MrPoniator: If all else fails, become a bureaucrat!
Ear Worm: Say it with us! Winter Wrap Up! Winter Wrap Up!
It later turns out not even Discord can resist it.
We also get "Fruit bat roundup, fruit bat roundup!" to the tune of this song in season four. The production team definitely knows what their best song is.
Why was Twilight having such a tough time getting her saddle on in the Cold Open? She'd probably been using her magic to help her do stuff like that for years, and this was the first time she'd tried doing it without.
"She's awful strong for such a little pony"...and a unicorn mare. All the other ponies pushing the snow plows are Earth pony stallions. Word of God states that, according to the magical rules of Equestria, Earth ponies have greater physical strength than unicorns or pegasi, and stallions are typically bigger and stronger than mares. Applejack has every reason to wonder at Twilight's ability to even move a snow plow, let alone move it faster than the stallions.
Before it was ever stated outright, moments like this are way fans have always assumed strength to be an earth pony advantage. Applejack isn't actually bigger than Twilight, and Applejack can move the plow just fine (we don't see it this episode, but we do see it.) Twi would have to be bigger than AJ to do the same work, for some reason - and then you notice there's not a single unicorn or pegasus, not even one, among the plowing team. It's not stated outright 'til the end of season four (Celestia makes it clear that strength is part of the package earth ponies have for tending the land, which is what can be taken from them by the magic-stealing Big Bad who started with unicorns but won't be stopping with them.) but it's been visible as early as this episode, number eleven.
Why have the ponies of Ponyville always been late for spring? Maybe it's the song that they insist on singing year after year. They just can't concentrate with that catchy tune floating around in their heads.
Harsher in Hindsight: Possibly, AJ was so furious with Twilight not so much because Twilight made a huge mess, nor exactly because Twilight used magic, but because Twilight didn't see how culturally disrespectful that was. The Earth ponies who founded Ponyville were Applejack's ancestors, including Granny Smith who AJ clearly adores. Earth ponies don't get nearly as much love from fans as pegasi and unicorns do, due to their natural powers being more Mundane Made AwesomeSuper StrengthGreen Thumb abilities than Inexplicably AwesomeColor Coded Magic or Flight). Twilight's use of magic unintentionally disrespected the Apple family, particularly Granny Smith, and by extension, AJ herself. Twilight must have realized it, too - that's why she got so upset.
Besides, there's no telling how many times over the years AJ and everyone in the family has had to defend the tradition from all types of ponies.
Memetic Mutation: Rainbow Dash's mention of a pegasus named Ditzy Doo who screwed up the bird migration last year got many fans suspecting this was actually Derpy Hooves' real name, and it remains a popular choice for people who find her fan name offensive.note According to Lauren Faust, who at the time represented Word of God for the series, the gray pegasus was intended to be Ditzy Doo, but "she can be Derpy if everyone likes".
Moe: Twilight actually displays more of it in this episode than any other... even more than Fluttershy. Especially her tearful look after Applejack scolds her...