Databases design decision - Mobile & Web
You can consider the web and mobile as just delivery interfaces to the functionality/services for your application. With this regard the backend functionality (server side - processing etc), should be the same for both interfaces, so yes they should have the same database. 

You may want to add data to identify the sources of data addition or changes or traffic, but essentially its the same app. 

UPDATE:
Think of a mobile app just like a separate UI to be supported, I am not going to exhaust the possibilities but look at potential list of possible interfaces below:


Web Application - available 
Mobile App - is this HTML 5 web app, Android or
IOS? each may require its own 'different infrastructure'
Desktop app - for a subset, administration, operations 
Web Services (REST, SOAP) - for machine to machine communication which does not have a definite UI per se, but may be provided to 3rd parties to develop their own apps (even the 3 UIs above can be considered as such) 


The way to think about it is that there is one application with a single database which is being accessed through multiple channels, whether from the same server or different servers