Say you create/edit a note in the OneNote app of Windows 10 PC, and the OneNote app of Windows 10 Mobile is closed (or open but in the background), then when you open the app in Windows 10 Mobile, are the new changes already there? Do notes sync in the background? Or they only start syncing when the app is opened, and not before? Same question for the To-Do app (adding, deleting, editing to-dos) in Windows 10 Mobile.
It actually syncs in real time with my Lenovo X1 Carbon, Surface Pro 4 and Apple Ipad 12.9 (2nd Generation) in OneNote. There is a slight delay, more noticeable on the iPad. On the X1 Carbon and Surface Pro 4 its only a few seconds but both are running the same operating system and same version of the app. If I have the same note open and do something like a copy and paste on my x1 carbon, and a few seconds later that paste appears on my Surface Pro 4, and about 10 seconds after that on my ipad. Its pretty neat feature. I am not sure about Microsoft To-Do but you could just pull down and sync it.