Disclaimer: I worked as an aerospace engineer for 15 years for the USAF. Our organization managed the 53 Federal Stock Group (1) (among others), which includes Bolts, Screws, etc. By this I mean to suggest I have some (dusty) knowledge of this subject.
While the quality control is very much a part of the process, as suggested, that alone is not the only difference. How the bolt is produced (the manufacturing process used) is also key.
So, for example, it is common to have bolts with "rolled threads".
Quoting from some industry literature (2):
Often, rolled threads are required by design because of their superior
tensile, shear, and fatigue strength. Other processes remove material
to produce the thread form, but thread rolling displaces the material
with hardened steel dies...
The result of moving the material grains (molecules) into the shape of
the thread rather than weakening it by removing material, is that the
grains become denser at the critical parts of the thread, especially
in the root and on the flank below the pitch diameter...
Figure 1: non-rolled thread
So, looking at the "not rolled" thread (fig. 1), one can see that threads which are cut (for example on a lathe) have exposed grain boundaries. Such a condition is undesirable for certain applications, for example under high impact loads. These areas would be prone to fracture/failure.
Compare the above illustration to the "rolled thread" illustration:
Figure 2: rolled thread
Here (Fig.2), as explained above, one can see that the grain boundaries flow around the contours of the thread, alleviating the exposed grain boundary condition and thereby producing a superior thread.
So, in conclusion, it is both the quality control requirements and the manufacturing process used to produce the bolt which make it aircraft quality.
FEDERAL SUPPLY CLASSIFICATION Groups and Classes (PDF)
Horst Engineering: Thread Rolling