When there are value adding (user and business) cases for people being able to use the app offline. Web is catching up with being able to give you these capabilities, but either way complexity of implementation (as well as UX) goes up with creating these.
Be really critical as to whether you really would need such capabilities. To give you an example, there is no reason why Instagram (a multi billion dollar business) shouldn't be able to 'post' a photo while I'm offline to some sort of a local storage and sync that up to the web once the app comes back online. But it doesn't do that. They basically haven't felt it's a strong enough use case for them. I think it is, but there you are.