The big button on the Home tab of the Picture Keeper app sometimes says Start Backup and sometimes says Resume Backup. They do the same thing, so tap whichever one is showing.
Both buttons run the same backup
Either button copies the photos and videos the app has not backed up yet. The app keeps its own record of what it has already copied — that record lives in the Picture Keeper app on your phone or tablet, not on the drive — so in normal use both buttons pick up where the last backup left off, and nothing that is already on the drive is copied again. (There is one exception, explained below.)
Why the label changes
The app shows Resume Backup once at least one photo or video has ever finished copying on that phone or tablet, and Start Backup until then. It goes back to Start Backup if you reset or erase the drive, or if you remove the app from your phone and put it back on. It also always shows Start Backup if you have turned on a clone backup — a fresh, complete copy that deliberately ignores the earlier backup history, which you would have chosen yourself in the app. Stopping a backup part-way does not change the label; the next backup still picks up where it stopped.
When it really does start over
If you reset or erase the drive, it is empty, so naturally the next backup copies everything again. The same thing happens if you remove the Picture Keeper app from your phone and put it back on: the app's record of what it has backed up is deleted along with it, so the button reads Start Backup even though your drive already holds a backup, and the next backup copies everything again. Nothing is lost and nothing is doubled up — the app simply puts an identical copy of each photo on top of the one already on the drive. It just takes longer than usual.
For the short answer on picking up a stopped backup, see How do I resume my last backup?; for the step-by-step version, see Resume a Backup Where I Left Off.
If you need any additional assistance, please contact Picture Keeper Support.