For example: Instagram does auto sync of new posts, but it also has a refresh functionality. So I never know if it is up to date and I must refresh. When you know about a manual refresh functionality the auto update is totally useles. You're never sure if the app tries to stay up to date or does only occasionally sync to show some (not the latest) updates.
Another example is google mail, it does auto sync pretty well and I always see the mails when they arrive. But the refresh buttons makes me suspicious if it's really up to date (in my experience it always is).