If you're mapping volumes to your application through VMFS, then you should always choose vCenter Synchronisation. Microsoft VSS Synchronisation can only be used when the volumes are directly mounted to the Windows server without any virtualisation software in the middle.
In your example, as you have a .vmdk mounted through VMFS then you should use vCenter. vCenter will use VMTools within the VM to engage with VSS (if the application has VSS components) to quiesce down any applications that are VSS aware.
I don't believe MySQL has native VSS components, so you may have to use VMTools with manual pre-freeze and post-thaw scripts if you wish to have application consistent snapshots.