Use the Trash button to delete single images from Photo Stream
To delete images from Photo Stream, you’ll have to pay a visit to the Photos app. Once there, tap the Photo Stream tab, select an image, and then tap the trash button. Alternately, you can tap the Sharing button before selecting an image, and then choose multiple photos to delete at the same time.
Deleting images you from Photo Stream on an iOS device, however, doesn’t work exactly as you expect if they were captured before upgrading to iOS 5.1 — meaning the process deletes images only on the device you’re currently using instead of removing them from Photo Stream on all your devices. That, by the way, doesn’t feel very Photo Streamy to me.
Use the Share button to delete multiple images from Photo Stream
You can delete Photo Stream images across all your devices on your Mac, too, from iPhoto ‘11 9.2.2. Once there, you can view your Photo Stream image collection and delete the pics you don’t want, and they’ll disappear from your iOS devices, too. Just like your iOS devices, images added prior to iOS 5.1 and iPhoto 9.2.2 will delete only from your Mac.
Now back to why you need to use Photos instead of iPhoto on your iPhone or iPad to manage images. My best guess is that Apple is actually eating its own dog food by restricting iPhoto to image editing, just as it does with third party apps: they don’t get to manage your photo library, either.
[Updated with even more details about deleting pics from Photo Stream, and some clarification, too.]