Why does deleted Photos show up again on my iPhone?

There is no option to delete photos from the iPhone and still have new photos sync with iCloud.

iCloud Photos is a syncing service. Any photo you delete from a device where iCloud Photos is turned on will also be deleted from iCloud and from any other device connected to iCloud Photos. You can potentially reduce the storage space required on a device by turning on Optimize Storage. See the following Apple article for more information: Set up and use iCloud Photos - Apple Support

Apple does not currently offer cloud-only storage for photos. If that is what you want or need, you may want to explore 3rd party options such as Google Photos or Dropbox.

The primary point of iCloud Photos is syncing photos among two or more devices (iPhone, iPad, MacBook, maybe iMac) and perhaps sharing them with other people. Secondarily, you can save some storage space on devices with limited storage (perhaps your iPhone) by enabling the "Optimize [device] Storage" option. A third benefit could be protection against loss of photos if the phone is destroyed and must be replaced (but not against accidental erasure of individual photos by the user).

The "Optimize..." option acts automatically when available storage on the device gets low by removing full resolution copies of some photos but keeping lower resolution copies. Your device has as many photos and videos on it when you look at the various catalog views, but it may need to download the full resolution copy if you want to edit a photo to view details of the photo.