Live migration depends upon the same hardware instruction set to be used on both systems. If using different version processors from the same manufacturer (Intel to Intel and AMD to AMD), it is possible to limit the instruction set used by specifying that the VMs be run in processor compatibility mode. This ensures a base instruction set is used for both. This is required for live migration because the actual memory of the running VM is transferred to the second machine.
Storage migration is not concerned with transferring memory - just storage. However, if transferring the system disk from one system to another, and you are employing live migration at the same time, then you have the constraints of live migration. But if you are just transferring the location of a data disk from one location to another without moving the running VM, you don't have to worry about processor compatibility.
https://social.technet.microsoft.com/Forums/windowsserver/en-US/e17cdb0e-150d-4486-a873-34c56ceea848/difference-between-quick-and-live-migration from the old forums provides a good comparison between live and quick migration.