The functional difference is that one means “installing from anywhere” and the other means “installing from outside the default app store”. They are different words with different meanings, one being more specific than the other.
If the outcome is the same, then there is no functional difference. If I say I need to see a doctor, there is a functional difference between a neurosurgeon and another medical professional. If I say I want to download a calculator app, there is no functional difference if I download it from a first-party app store or a third-party app store. You're splitting hairs. Stop supporting corporate hegemony.
If there is a functional difference, then why can't you say what the difference is? You continue to refuse to do so BECAUSE THERE IS NO FUNCTIONAL DIFFERENCE. All you're doing is installing an app. The only thing that changes is where the app comes from and if your corporate overlords approve of it's origin. So again, what fundamental difference does it make where the app comes from?
I'm saying they're the same. You're saying these two things that are functionally identical are fundamentally different. In what conceivable way am I splitting hairs?