Noel, when capitalized is another name for Christmas.
When it is lowercase it is a synonym for a Christmas carol. The original French word has the accent mark over the e, noël. In French Noël can also mean Christmas. So if you want to reference the French in your work, spell it this way.
A yule log is a Christmas dessert that is also known as a bûche de Noël.
The Christmas carol The First Noel is about the first carol sung at the first Christmas.
In Middle English the spelling of noel was nowell. Currently it can be spelled with one or two ells. Unless your audience is very familiar with the old style of spelling, it is best to stay with noel.
Another definition of nowel is the piece of a mold in the center which will make the final product hollow, or the lowermost part of a mold. This definition carries two pronunciations, either (no el) or (newel).
From now until New Year’s, come have a celebratory sip with Lavoie’s seasonal selections — the Noel Noir (literally “black Christmas”) martini, taking its name from Grey Goose Cherry Noir vodka and featuring the deeply dramatic Filthy black cherry; and the St. Nick’s Sangria, which —surprise! — “isn’t really sangria,” Lavoie says, because there’s no red wine. [Palm Beach Post]
It gifts you the perfect antidote for the Whitman Sampler of bah, humbug Noel platitudes. [Huffington Post]
“. . . we thought that we ought to retaliate in the same way, so we sang ‘The First Nowell,’ and when we finished that they all began clapping,” Williams wrote in a letter published in Truce, noting the two sides sang a carol together. [Toronto Star]