They do different things. Exactly how they differ depends on the settings on your previous device.
Roughly:
"Restore from iCloud" restores from a previous backup kept in iCloud. That would include a number of settings, the names and arrangement of apps, the siloed data of most apps (such as game progress, perhaps, or health data), and other types of data not actively synced through iCloud. It would include photos ONLY if you were NOT using iCloud Photo Library.
"Set up as new" skips the restore, cleans out the phone storage, and runs you through the initial setup prompts.
In either case, any data actively synced through iCloud would then filter down to the device. Examples would include contacts (if synced through iCloud), calendars (those stored in iCloud), photos (ONLY if you WERE using iCloud Photo Library), and iCloud Drive files.