Now that app based door lock replacements are getting more popular in the market, I'm wondering what the security risks are and why or why not this is more secure than an old fashioned lock with a physical key.
I'd argue that they are fundamentally no more secure than a traditional lock. These types of locks can also be opened by a standard physical key, so they by definition are susceptible to all of the weaknesses of a traditional lock plus any weaknesses introduced by the electronic layer.
Whether or not these additional weaknesses are significant is an open question, and will depend on the specific manufacturer and model. With that said, it's my strong suspicion that the traditional lock manufacturers who have begun building these digital models likely lack the software engineering experience and deep cryptographic understanding necessary to design and implement such a complicated product in a way that would withstand a skilled and determined adversary.
While @raz is correct that a door is only as secure as its frame, an important feature of a lock is to ensure that a break-in leaves physical evidence. Someone entering a locked room is not necessarily a thief just looking to ransack the place for loot. My biggest concern with internet-connected locks is that a software flaw would plausibly allow attackers to silently bypass locks of their choosing, leaving no physical evidence.
That said, if I had the option I would consider putting one on my home. The threat model there is, generally, a thief looking to ransack my home for expensive physical goods, and any additional vulnerabilities introduced by the internet layer are unlikely to be exploited by this type of attacker over simply using a crowbar on the door or breaking a window. The added convenience may be worth the risk. On the other hand, I would likely not choose to install such a lock on the door to a rack of servers, the door to an office suite, or to a safe containing valuable documents.