In a nutshell, shipping boxes and moving boxes differ along a few major areas – design purpose, reuse, weight capability, structure and even packing tape used. Similarities include but are not limited to the fact that both types of boxes are made from corrugated fiberboard and can be bought at any packaging supplier including the plethora of online retailers.
Corrugated fiberboard is the building block of cardboard boxes. It is a sandwich of containerboard paper (a thicker heavy paper) and flutes. The flutes are the wavy corrugate in between the walls of a large cardboard box. They can be of various thickness and rating and give sturdiness and strength of the shipping/moving container. In addition, they increase the stacking ability of a certain type of box plus they are the most significant factor in parameters such as puncture resistance or edge crush test. These are essential when choosing a box according to product specifications. In addition, the flutes provide some protection and weather proofing within a reason which could be crucial as related to food spoilage and preservation. To complete the box an inside and an outside liner are added to the fiberboard sandwich. The latter can have one or more sets of wavy flutes according to desired strength and conformity to product requirements.
The corrugated industry is no modern marvel and follows rules of economics such as supply and demand. Once a demand for a box has been identified, container of the proper size and characteristics would be made and put in production.
In conclusion, using almost any cardboard container for moving would be acceptable since even brutal movers tend to only handle the moving boxes short term. On the other hand, shipping boxes have to be new, designed to product specification and packed and mailed with the understanding that they will change hands many times and certain free falls and drops are to be expected before they reach their destination.