Dan,
iniPhoto you would see the filename in the title field of a photo, if you did not assign a custom title.
Photos does not show the filename as the default title. You can only see the filename in the Info panel and copy it to the title field, if you want to use it as the title.
To see the titles below the thumbnails in the browser enable the option "View > Metadata > Titles" in the view menu.
I posted a script to copy the filenames to the title field for selected photos, to make it easier to copy it for several photos at once, see:
Photos for Mac: Batch Changing the Titles to the Filename
If you are using iCloud Photo Library, try the version 2 of the script.
But anyway, the update to MAcOS X 10.10.5 is causing trouble with the scripting, and there are several reports, that not all scripts on https://discussions.apple.com/community/mac_os/photos_osx/content?filterID=conte ntstatus%5Bpublished%5D~objecttype~objecttype%5Bdocument%5D
are working well any longer, so don't be disappointed, if you encounter errors when you try the scripts.