AMI of Running Instance versus AMI of Shut Down Instance

In general, it's better to take a snapshot of a shutdown instance (rather than a running one) if possible.

The reason is that if the instance is running then there may be files which the operating system has open which are not completely written to disk when the snapshot happens. Consider a database engine that is writing into many tables and indexes - it's quite possible that at the exact moment the snapshot is taken of the disk that there is data still in memory. When that snapshot is restored, the database will be inconsistent and this could easily lead to data loss.

Is this always the case? No. But it would be difficult to predict when a server is not writing to disk. Some applications (databases, specifically but others too) have flags to "quiesce" operations - that is, pause things while a snapshot is taken which is much better than taking a snapshot of a "live" disk.

Edit (adding): If the instance is hibernated then it is effectively paused. But that doesn't mean that the data has been written to disk successfully. If the applications are tracking the operating system state and are able to receive (and appropriately process) signals from the operating system about the hibernation then they should write data to disk. But that's not a guarantee and it depends on what is running on the instance.