Taking a snapshot when a virtual machine is offline consists in creating a "delta" virtual disk, attached to the original one, in which all new data will be stored. This way, initial data will remain untouched. When the machine in online, you do not only create that delta disk but also store the content of the memory. You can then restart the VM in the exact same situation as when it was running during the snapshot.
It's always better to have a snapshot when the guest OS in offline. You are then absolutely sure that the content of the disk is consistent. But if you can't stop the machine due to production constraints, an online snapshot is the only way to have an instant backup.
When you configure a virtual disk as independent, you inform ESX not to generate any delta disk for this vmdk in case a snapshot is taken for this VM. Persistent mode means that all data is written to the disk when the guest OS is running. Non persistent means that a new temporary "delta" disk is created on every reboot of the VM and this temporary disk is erased on shutdown. In this case, the machine always restarts with exactly the same data on disk.