If your product is never downloaded, and you're constantly serving your clients the latest version, it's not a huge issue for on a user-level. It would help with debugging, if the user happens to use an older copy of your app and you need to know exactly which version it is.
But it helps immensely on a developer-level. It allows you to keep more sensible track of your applications progress with the semver structure. Your devs will know that 1.0 software won't be compatible (or will have breaking changes) with 2.0+ software. Or you'll know if an update was more of a bug fix than a feature.
Not sure what you're doing now, but it's probably already some derivative of semver. By adopting a more standard policy, you create less confusion when new developers have to adopt your codebase (oh yeah, we're like semver, but do XYZ1298485 different).