How can I stop my photo stream from sharing with my other devices?

To fix this you need to be on a separate iCloud account (when you share and account, you share your photo stream). To migrate your phone to a separate account, start by saving any photos not in your camera roll to your camera roll by opening your my photo stream album, tapping Select, tap the photos, tap the share icon (box with upward pointing arrow), then tapping Save to Camera Roll.

Then go to Settings>iCloud, tap Delete Account, choose Keep on My iPhone when prompted, then sign back in with a separate Apple ID to create your new account and choose Merge to upload your data. (Deleting the account only deletes it from your phone, not from iCloud. Your mom's phone wont be effected by this. Also, you can continue to share the same ID for iTunes and App Store purchasing, it doesn't need to be the same as your iCloud ID.) Once you are on a different account, your photo stream will be no longer be merged with your mother's.

Yes. Once you are on separate iCloud accounts you will no longer get each other's photo stream photos. You can continue to keep the same ID in Settings>iTunes & App Store and share the same music and other purchased media.