We migrated earlier this year and had a similar problem. Outlook (classic) works differently than the OWA version. They keep the classic version so people don't spontaneously throw a chair out a window. It's being phased out slowly.
They also keep their own inboxes; emails downloaded to or sent from the old version are not visible on the new version.
I'm betting in 15 years, people will still be using Outlook (classic). This is the culture.