What are the advantages of going with an LDS publisher vs. a national publisher?
First, if your book has LDS references in it, an LDS publisher is going to be more open to that than your average national publisher, so your chance of getting it accepted increases.
If your book is specifically targeted to an LDS audience, an LDS publisher is going to hit that target market on the head. They advertise straight to that market so every dollar spent is hitting your intended reader. A good portion of a national publisher's marketing dollar (assuming they pick up your manuscript in the first place) would be "wasted" on uninterested readers.
There's the reader trust factor. Readers who specifically want books with LDS content are going to look first for books published by an LDS publisher. They trust those publishers to provide content they are comfortable with, whereas a national publisher might slip in language or scenes that make them uncomfortable.
A sense of LDS community and loyalty is also a big deal. Some LDS readers will read/purchase books by LDS authors/publishers simply because they want to encourage LDS authors. I have to admit that I read Shannon Hale for the first time solely based on the fact that someone told me she was LDS.