It really depends on how you have configured Photos. Did you decide to enable iCloud Photo Library, or are you just using the Photos app the same way you used iPhoto (meaning, are you still importing your photos to the Photos app from your phone and not using iCloud Photo library)?
Also, just like with iPhoto, if you are not using iCloud Photo Library, and you delete a photo from Photo Stream, it deletes it from the Photo Stream on all devices that are signed into that same iCloud account with Photo Stream turned on. However, if you delete it from the Camera Roll, it only deletes it from the Camera Roll on that device.
If you do not use iCloud Photo Library, the Photos app works in a very similar manner to iPhoto in terms of importing and sharing photos. If you do use iCoud Photo Library, the entire process changes dramatically. Here are a couple of support articles on iCloud Photo Library that may clear up how this functionality is different from just using the Photos app without iCloud Photo Library:
iCloud Photo Library FAQ - Apple Support
iCloud Photo Library Help - Apple Support
I use Photos, but I also still use iPhoto and Aperture. I also do not use iCloud Photo Library. The settings for automatic import, Photo Stream, and iCloud Photo Sharing in Preference in the Photos app is very similar to those same settings in iPhoto. I have found that only one of them can be turned on with these settings (if you turn both on, the last one turned on is the one that stays on, and the other one gets turned off), but all I want to make sure of is that I do have my Photo Stream automatically importing. I can then manually import those photos to the app that does not have those preferences turned on.
Best of luck,
GB
