Or are they different in other respects also.
Not a lot of difference, 5.56 are usually surplus military and have thicker case walls, that changes the internal volume of air space and produces hotter loads. Not allways the case but most of the time.
Shoulder of the 5.56 is slightly different and will not seal a 223 chamber correctly so there is a posibilty of gas coming back to the bolt and the shooters face. That is why they say you can use a 223 in a 5.56 barrel but not to use a 5.56 in a 223 barrel. However, if you have resized the 5.56 case with 223 dies and did it correctly, you should have a 223 case now that is marked 5.56. One of lifes little querks.