Ha, I've just finished writing a lecture on Christmas music. To answer your question, I'd note that Christmas carols are simply a subset of Christmas related music. As a matter of fact, the song you’re referring to, Adeste Fideles, was not originally meant to be a Christmas Carol, it predates them. It’s actually a Catholic hymn, more often than not attributed to King João IV of Portugal. The current music you know as Oh Come All Ye Faithful was written by English Catholic composer and Catholic hymnist John Francis Wade in the XVIII century CE (I will absolutely not get into the whole Jacobite connection because, while a good enough theory, it still needs a lot of substantiation). That being said, we have some very early examples we can look at.

The oldest Christmas religious hymn of which we have records is Veni redemptor gentium, attributed to Saint Ambrose of Milan, and it dates to the IV century CE. The hymn was arranged for a chorale setting during the Protestant Reformation period, and in its current form, it draws on a melody written by composer Michael Praetorius around 1605, called Puer Nobis Nascitur, which can be found in one of Praetorius largest compilations, the Musae Sioniae. Here's a gorgeous rendition by Riga's Schola Cantorum.

If we move forward several centuries, we get to Resonet in laudibus. We don't really know who wrote it, but it's at least as old as 1360, when it was published in the Moosburg Gradual, an English songbook and compilation work. Nowadays, we tend to listen to its musical form as composed by Orlande de Lassus, published in 1569 as part of his compilation Cantiones aliquot cinque vocum. You can hear it here by The King's Singers.

Back to the XVIII century, we find a 56 year old Georg Friederich Händel composing, in 1741, one of the most renowned Christmas choral works in history: Messiah. He composed it in only 24 days, following the libretto and instructions that his friend and patron Charles Jennens had provided. Messiah is a behemoth: between two and a half and three hours long, orchestrated for a full choral/orchestral ensemble, it spans three different movements detailing the canonized version of the life of Jesus Christ. The first movement narrates from Isaiah’s prophecy and the annunciation, through the Immaculate Conception, until his birth and the flight into Egypt; the second movement delves into the Passion, the trial, crucifixion, resurrection and ascension, and the final movement goes into Paul’s teachings about Jesus’ holy place in heaven. Messiah is so long, its movements are usually performed alone, with the first one, colloquially called Christmas, tends to be the most commonly performed one. One of the songs in this movement is For Unto Us A Child Is Born, which continues to be a popular Christmas song in the English speaking world. Here it is sung by the Tenebrae choir with the LSO and Sir Colin Davis.

Finally, for a proper Christmas carol, yet another very, very pre-1900 one is The Twelve Days of Christmas! Many people don’t know this, but it was first published around 1780 in an English children’s book called Mirth without Mischief, which is why the lyrics, usually considered to be of French origin, come from a nursery rhyme. The idea of the Twelve Days comes from the traditional Christian liturgical celebration of Twelvetide, established by the Council of Tours of 567 for the celebration of Christ’s birth, beginning on December 24th and culminating on January 5th, during the feast of the Epiphany. This might be my favorite version of the song, by Spanish tenor Plácido Domingo and Norwegian soprano Sissel Kyrkjebø, for the 1994 Christmas in Vienna season.