One travels and the other does not, and the reason is what the nitrogen is doing inside it.
The Construction Difference
A cryoshipper holds liquid nitrogen absorbed into a porous inner matrix rather than as free liquid. Samples travel in the vapour phase and the vessel can be inverted without spilling. A storage dewar holds free liquid, is designed to sit still in a controlled room, and is not a transport vessel however carefully handled.
Why That Determines the Paperwork
Because the nitrogen cannot escape in any orientation, the shipper itself travels as not restricted under Special Provision A152 of the IATA Dangerous Goods Regulations, provided the packaging cannot build pressure and the air waybill is annotated. Packing Instruction 650 governs the biological samples inside where those are UN 3373 Category B. It applies to the samples, not to the nitrogen, and the two are conflated constantly.
Practical Consequence
Hold time is finite and falls with every opening, orientation change and rise in ambient temperature. A shipper is a transport vessel with a clock running, not a portable store. Compare against storage vessels and see the CryoStork range.