There is no significant difference between the wording and requirements of parts 23 and 25, the only real difference is the level of rigour required by part 25 in the assessment, which is considerably greater. Also however, because weight is usually more customer-critical on part 25 aircraft they are usually designed as close as possible to the margins, whilst they may be larger on part 23 aircraft.
The 1.5 safety factor in paragraph 303 is the backstop requirement for the margin between limit loads (that which may be seen in service) and ultimate loads (what it's supposed to take for 3 seconds without failing for 3 second, but a failure after 4 seconds is acceptable in the standards). Other margins may be added in - for example for bolted joints, composite materials, loaded hinges - but the basic principle is always there.
And the principle is only partly related to either speed or normal acceleration. The safety factor is applied to the actual loads in the structure - which are inevitably complex and dependent upon multiple factors.
So within a wing, for example - there are loads in the mainspar due to weight and Nz, but these will be modified also by speed (because of drag and torsion loads acting aerodynamically), and also will be affected by the alleviating weight distribution of the fuel within the wing tanks - increased fuel in the tanks, particularly outboard, will decrease the stresses on the wing structure.
At high speed, the most significant loads on the wing are actually torsional, and yes the 1.5 factor is applied to the predicted torsional loads. But they are not linear with speed - they're roughly proportional to the square of speed. So, the simplistic factor is SQRT(1.5)=1.22. In practice there are more players than that, which include potential for instability, downwash, balancing and torsion loads on the tailplane (which will also be dependent upon CG position).
The only thing that you can really rely upon in either a part 25 or part 23 aeroplane, is that when the aeroplane was flight tested, it was taken to Vdf and Mdf - flight test diving speeds; above that for whatever criteria were used by the flight test and airworthiness teams, it was deemed unsafe to fly.
Vne is set not above 0.9Vdf, and Mne is set not above 0.9Mdf.
1/0.9=1.11 or 111%.
So basically your actual speed margin (in theory at-least) is another 11% on speed.
But that was a flight test team, in a new and instrumented aeroplane - you don't have either of those things to your favour.
So, there is a region between maximum speed and 11% above that which can be labelled "here live dragons", but until 1.11Vne (or 1.11Mne) you don't have a guarantee that the dragons will eat you. Above that, you can probably rely upon that happening.