What your role is mainly depends on what you’re hired for. A web developer can be a perfectly competent software developer. A bit harder for the other way around but it’s definitely not a huge jump. One thing I’ve found that is difficult for most developers is systems admintration. You’re often called on to do things in that realm but it’s outside the skillset you usually train for. If you want to be more useful to an organization, don’t try to be a web developer and a software developer, try to be a systems admin and just one kind of developer.
Good luck!