The Children of Húrin contains some content that is new, though most of the "new" parts are very similar to previously published material.
There is an appendix published in The Children of Húrin called "The Composition of the Text", where Christopher Tolkien describes where the text in the book comes from, and how it differs from previously published material.
[In Unfinished Tales] I omitted a number of substantial passages (and one of them very long) where the Narn text and that in the much briefer version in The Silmarillion are very similar, or where I decided that no distinctive ‘long’ text could be provided.
The form of the Narn in this book therefore differs in a number of ways from that in Unfinished Tales, some of them deriving from the far more thorough study of the formidable complex of manuscripts that I made after that book was published. This led me to different conclusions about the relations and sequence of some of the texts, chiefly in the extremely confusing evolution of the legend in the period of ‘Túrin among the Outlaws’.
The Children of Húrin - "Appendix: The Composition of the Text"
It is not a very long appendix and I would recommend you read it in full to get the answer, as it is probably the best source. Quoting the entire appendix would be beyond the scope of this answer, but to summarize some of the points:
The text of The Children of Húrin is for the most part composed of text of "Narn i Chîn Húrin" (abbreviated as "Narn") which was also the primary source for the material included in Unfinished Tales.
In some places passages from the Narn were skipped in Unfinished Tales because they were too similar to the shorter "Silmarillion version" Tolkien wrote, which was already published by Christopher in the Silmarillion.
In some places there was no shorter "Silmarillion version", and so Christopher created one for the Silmarillion using the text of the Narn, but because that version he created and already published was therefore too similar to the Narn, the longer versions were skipped in Unfinished Tales.
In some places the original Narn material was very rough, and Christopher took some editorial freedom in Unfinished Tales that he later felt was unnecessary, and so the version in The Children of Húrin represents a different, more accurate reading of the original manuscripts.
In some places there are actual gaps in the Narn material, and Christopher filled these the same way he created the Silmarillion text (e.g. from the Annals of Beleriand)
In some places there are just minor errors or inconsistencies with some of the source material and previous published versions that Christopher has corrected.