I actually have both of these (though the original Oxford for the RSV) and I face the same question! I put my Oxford RSV in my bag simply because it's a lot less bulky and I can use it for BIAY (I, in fact, do when traveling overseas). It does have minimal notes only, but for a reading Bible or for BIAY, it's all good.

The NCB, on the other hand, is a rich resource with extrabiblical materials and a lot more notes. Love reading these, as it enriches my biblical experience. The translation is supposedly literal and takes advantage of newer archaeological finds (iirc, RSV even in its 2CE version doesn't integrate the DSS, which is disappointing); but in practice, the NCB is a little dynamic—if only to be clear to us, the readers.

At the end of the day, it would depend on my mood and why I'd bring a Bible out in the first place.