The NAS1303-NAS1320 specification is one spec covering multiple sizes (diameters) of the same bolt. NAS1304 is for the 1/4" dia bolts. Same with the NAS6603-NAS6620 spec.
The last revision of NAS1303-NAS1320 states, "Inactive for new design after Oct. 30, 1981. See NAS6603 thru 6620."
What this means is that while it's recommended that no new designs specify the NAS13XX, there's nothing wrong with the bolt itself and there are likely still bunches on shelves (but probably none in production anywhere if no one is using them for design). Bolts don't get old sitting on a shelf.
Looking at both specs side by side, there are no huge dimensional or finish differences between the two bolts. Metallurgy is the same, with one additional steel alloy (4140) possible to use for the NAS6604 which wasn't permitted for the NAS1304. Other possible alloys include 4340 and 8740 for both bolts. Heat treat and final tensile properties are identical for both. Both are 100% mag particle inspected at this size.
Bob is correct above - for some diameters, there are slight length changes in the threaded portion. For the 1/4" bolts though, they're the same.
Not advising you, but I'd have no problem using either one interchangeably.