Answer:
Hymns are usually religious, taking inspiration from the Bible and acknowledging God. Christmas Hymns usually have lyrics associated with the birth of Christ. For example, "Mary's Boy Child", "For Unto Us A Child Is Born", "Away In A Manger".
Carols are associated with Christmas but not necessarily religious in nature. For example, "Jingle Bells" and "Rudolf the Red Nose Reindeer).
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I was taught that in the time of the origin of these songs, the church hymns were always sung in Latin but Carols were sung locally and in the local dialect.