While some of the arguments in favour of mobile applications apply in many cases, I'd like to consider a specific recent example: The GitHub mobile app.
I have been using GitHub on mobile via its web application for years. It worked fine for the most part though some feature (such as pull requests) weren't as easily accessible on mobile as they are on desktop machines.
However, with the advent of their new mobile app, not only is its UX better when compared with the web app (e.g. when reviewing pull requests) but due to the banner notification reminding you to use the mobile app when accessing GitHub via the web app on mobile devices the UX of the web app to some extent even has decreased.
The thing is, there are no features within the GitHub mobile app that actually warrant having such an app. Everything the app does would have been possible on the web and with mobile browser APIs, too.
So, in that specific case the purpose of a mobile application perhaps mostly comes down to having such a mobile application, i.e. it's more about marketing and distribution channels than it is about technology or design requirements.