It's a good example of a pre-internet (pre-telegraph, in fact) meme. Harriet Butler, a friend of the Moore family, saw the handwritten original of the poem on a table in the Moore house in New York City when she visited for Christmas in 1822, copied it down in her journal, then showed it to a friend, who also copied it and passed it on to the editor of the Troy Sentinel in New York State, who printed it on December 23, 1823. In a matter of days (around New Year's), two other newspapers had already reprinted it, and it quickly became an annual tradition in lots of publications.
Older Than They Think: The poem originally closed with "Happy Christmas to all, and to all a good night", and it's easy to assume the change to "Merry Christmas..." was a 20th century retcon to replace the outmoded original usage, but newspaper reprints with "Merry Christmas to all..." from 1829 and 1830 have been discovered.
The Rankin-Bass special:
Alternate Character Interpretation: It's commonly given that Santa never really intended to skip the town. After all, he still had presents ready. Though, granted, he would've had a year to prepare them...
Fair for Its Day: The Thomas family are a Token Minority family as the only black people in Junctionville, who don't even have any spoken lines. But they're the first black people to appear in a Rankin/Bass Christmas special, they appear prominently throughout the special, and their son Davy, "the best artist in school," gets a poignant moment in the limelight when he sadly drops the picture he drew of Santa into the sea.
Parody Displacement: The 2000 South Park episode "A Very Crappy Christmas" parodies this special's plot, with the lead human characters creating a public display (the Spirit of Christmas short film) to bolster Christmas cheer, non-human characters (Mr. Hankey the Christmas Poo and family) aiding the effort, and Kyle singing "Even A Miracle Needs A Hand", with his face even changing to Joshua Trundle's at one point. As Twas The Night Before Christmas hasn't been a regular sight on American TV during the holidays in recent years, the South Park episode has become better known to many fans, though the original special got a boost in airings on AMC in 2018.
The town's sadness when they think Santa isn't coming. Particularly the two sights Father Mouse takes Albert to see as proof of the harm his letter has caused: a children's hospital full of sick and injured kids crying, and the Token Minority boy Davy Thomas dropping a picture he drew of Santa into the ocean from a cliff.
The Trundles (and therefore the mice too) facing poverty and hunger when the town loses its faith in Joshua's clockmaking after the musical clock explodes. Followed by Albert crying Tears of Remorse as he admits that it's all his fault.
Father Mouse and Joshua both shedding Manly Tears as midnight arrives with no sign of the clock being fixed or Santa coming. Fortunately, the clock's song does start up just a few seconds late, and Santa does appear.
are only the tip of the iceberg. Now has its own trope: The Parody Before Christmas.