Even though there is overlap, (Thorne and Blandford) and (Arfen and Weber) are quite different books; they are not meant to do the same thing. Thorne and Blandford treats advanced Classical Physics. At times, it uses standard Mathematical Methods to do this, but the emphasis is on the physics. At other times, Thorne and Blandford uses more geometrical mathematics that isn't so standard in Mathematical Methods texts. Arfken and Weber has some application to physics, but emphasizes the methods.
Likes and dislikes are very personal and subjective. I am only lukewarm with respect to Arfken and Weber, but many folks really like it (including my wife!).
Most people probably want/need the mathematical techniques in Arfken and Weber more than they want/need Thorne and Blandford's treatment of advanced classical physics. A couple of months ago, my wife came to my office, saw Blandford and Thorne, read the title and subtitle, and exclaimed "What is THIS doing on your shelf!!!" She never would have predicted that I would buy such a book.
You nailed it pretty well, I'm more interested in the math right now as that's needed to understand the physics I want to relearn. Arfken chapters are more discrete in that you can skip around and I felt that Thorne's book was more sequential building up a base one chapter topic at a time. Perhaps, when I retire I'll get a copy with my final paycheck.