The most useful product offered by Nest that the Ring system does not have is a connected lock. This tamper‑proof, key‑free deadbolt connects to the Nest app, letting you lock and unlock your door remotely. Users can also create passcodes for family, guests and people they trust, or give them a Nest Tag programmed to let them in only at certain times of the day.
The Ring Doorbell Camera security system was acquired by Amazon. They currently offer doorbell cameras and exterior security lighting cameras. And according to their website product page, Ring will soon be adding indoor security and environmental protection products to their lineup as well. This is great news based on how popular their current products have become.
Update 1/24/16 - I have found a fantastic use for the Ring Doorbell and Chime. My mother-in-law, now in her 90s, lives by herself. She has become hard-of-hearing and often does not hear the doorbell. By installing the Ring on her door, she now is bothered much less (solicitors tend to recognize it and don't seem to like it) and when someone rings the bell, the Chime, placed by her favorite chair, gives her a better chance of hearing someone at the door.