Is it possible to launch a different windows store app when a user clicks on a button within my app?
Sadly, it is not possible as Win-RT apps are not allowed to launch other apps.
There's a workaround that allows you to open a specific app if it's registered to open with a specific URL:
http://msdn.microsoft.com/en-us/library/windowsphone/develop/jj206987(v=vs.105).aspx
I'm not sure if it will help much, but you can open the store and go to a specific store app page with:
Launcher.LaunchUriAsync(new Uri("ms-windows-store:PDP?PFN=" + storeID), a);
Where the storeID looks like '30416[AppName].[AppName]_94hq21bnosq1p' and can be obtained from the web store source code:
http://apps.microsoft.com/webpdp/en-gb/app/Fhotoroom/4d91298a-07cb-40fb-aecc-4cb5615d53c1