Comment 18 for bug 1066113

The only thing Variety does is call a gsettings command to instruct Gnome "look, this should be the new wallpaper". From there on Variety does nothing, Ubuntu/Gnome/Unity is responsible for redrawing the wallpaper and dealing with LightDM. LightDM has its own requirements about the image, one of them is that obviously it should be readable by everyone, there are others too and I don't know them all. For example an image in your Pictures folder would work, but not inside a subfolder in Pictures.
I'm pretty sure images on mounted partitions never worked, not even on 12.04, probably because the mount point will not necessarily be there on next reboot.

In any case, Variety does NOT deal directly with LightDM, it never has, not in 12.04, not in 13.04, Gnome does.

Why it is different between 12.04 and 13.04 I don't know, it may some difference in your setup, it may be due to updates to LightDM or something else in Unity/Gnome.

Now, if you set a specific wallpaper directly from Appearance and it appears in LightDM and then you set it from Variety and it does NOT appear in LightDM, this would prove Variety is not doing something right in this regard, and I'll be open to suggestions - everyone can edit the ~/.config/variety/scripts/set_wallpaper script and try what command would actually fix things with LightDM. But this hasn't happened on my machine.

For clock, filters and quotes Variety needs to build a new image, it has no other place to save it but the ~/.config/variety folder, LightDM cannot use it from there, so basically this is a known and unavoidable problem.