Unfortunately the vSphere Client GUI dosn't allow you to select snapshots as existing virtual disks and by starting the VM with the base disks, the CID (as well as some data) for the base disk was updated which results in the issue you see now. Depending on how long the VM was powered on in this state and the size of the snapshots it might be possible to relink the virtual disk files without much corruption in the guest OS.
What you may do is to enable SSH for the host (Configuration tab -> Security Profile -> Properties), then use WinSCP and download the files I mentioned before. For the .vmdk files, only the small descriptor files (a few hundred bytes in size) are important.
André