No there isn't a pressure difference and the one you are going to try and cite neither you nor your source understands what it's citing.
Further that higher pressure you want to claim 5.56 is is within tolerance of .223 Remington.
What is it with medium and large 2a accounts on her being completely uneducated on the actual workings of firearms.
.223 Remington is 5.56x45. they're the same thing and completely interchangeable.