The Apps icon on the iconbar is a read only pseudo filesystem (called ResourceFS) made up of links to stuff held elsewhere:
1. Resources and applications (Edit, Draw, Paint) held in the ROM image
2. Applications held in the Apps directory on the SD card
3. Anything added using the configuration tool (see below)
I don't have my Pi to hand so I'm not sure if NetSurf was added through option 2 or 3, but either way you need to replace the real copy on the SD card rather than try to update the link in ResourceFS.
To see what is added using 3, click with the Menu (middle usually!) mouse button on the Task Manager (Raspberry icon bottom right) and choose Configure. Click the Boot icon, then on "Add to Apps". If NetSurf is in here then this will show you where it is actually located.
While you are in the Boot section of Configure, note that there is an Install icon. NetSurf comes with a !Boot application that needs to be dropped into here to merge it with your current boot.
Similarly, going back to the main Configure window there is a System icon. Drop the !System supplied with NetSurf into here to merge that too.
It is ironic that NetSurf is one of the few RISC OS applications that would benefit from package management, but it isn't yet packaged