MailTips, out-of-office messages, and similar features understand that Office365 and onpremises users are part of the same organization.
Delivery reports and multi-mailbox search work with users who are onpremises and those working in ExchangeOnline.
Authentication headers are preserved during cross-premises mail flow, so all mail looks and feels like it is internal to the company (for example, recipient names resolve in the global address list).