Apps are just a different name for software and browsers are software to view content on the web. We will for sure see development of browsers and apps, especially towards multimedia and interactive use, but they will likely be evolutionary. Also there will be a certain amount of convergence as the possibilities to run code in the browser (e.g. through developments like PNACL  and asm.js) increase and local apps increasingly depending on and interacting with services in the cloud. 

Both the browser and the app are interfaces for interacting with the services available online, including internet viewing as one of the cases.

Right now a paradigm shift happening in the interaction interface (started in 2015). A substantial part of interactions will move from mobile apps with a GUI to conversational interfaces (also known as chatbots). This trend mainly is driven by the fact that users spend most of the time on mobile devices in messengers, in fact, more than in all other apps. So services need to move to where the users are, i.e. into messengers.

On the other hand, recent development in NLP allowed making the conversational interactions more user-friendly. Practically this means that the users do not need to formulate their intent in a command line-like strict style, but can use a regular language and the machine will still be able to understand what they mean. Such understanding makes it possible to interact with a computer in a declarative style expressing what the user want to accomplish (e.g., "cut out the third face from the left on a group photo"). Before, the user-computer (in app or browser) interaction was mainly procedural. Typically the user needed to execute specific steps in their particular order (e.g., select the face on the photo, press the cut button). There is an expectation that this will enable more common interaction for less tech-savvy users, particularly pertinent for the next 4B people still to join the Internet.

Answering your initial question, some chat apps already provide developers with a possibility of creating custom messages which can be thought of as small web pages. So instead of seeing the whole webpage or an app screen at once, a bot would present small parts of the web in a form of a message particularly relevant to the conversation state.

In summary, chatbots are coming, and they will force us to rethink and redesign many aspects of the human-computer interaction we are used to today. This includes interaction with online services currently available as web pages or mobile apps.

This post provides an extensive intro into the chatbots: https://medium.com/chat-bots/the-complete-beginner-s-guide-to-chatbots-8280b7b906ca#.axmbc4o50

Hope this helps.

Internet and Web go to meet the Internet of things and the Web of things