As I understand it,
imap is a protocol while Exchange is a mail server.
IMAP stand for Internet Messaging Access Protocol and is a commonly used alternative to POP3 (Post Office Protocol).
Microsoft Exchange Server is "a mail server and calendaring server developed by Microsoft".
imap, emails are synced and can be viewed in multiple clients. Copies of those emails remain on the server.
For the most part, yes.
imap, emails do remain on the server (until deleted) and copies are downloaded to individual clients. If a given client sends a message to perform an operation on a given email (e.g. move/delete), then the server typically performs that operation.
As a protocol,
imap "provides mechanisms for clients to detect changes made to the mailbox by other, concurrently connected, clients." However, clients can technically ignore those messages or not sync local changes (in practice, this applies most often to sent or deleted emails).
Exchange is the same, only that instead of just syncing emails, Exchange also syncs collaborative tools such as contacts and calendar.
For clarity, per your opening statement,
imap and Exchange are two completely different things. Furthermore, while current versions can use
imap, Exchange natively use a proprietary protocol called
mapi to communicate with clients.
That said, in the broadest sense, you are correct in that servers which implement
imap often only manage emails while Exchange servers deal with contacts and calendars by default (which
imap itself does not provide mechanisms for).
Also, sent items are stored locally with
imap while sent items are stored on the server for Exchange.
imap, while many clients will automatically store a copy of a sent email on the
imap server itself (which can then by synced to other clients), this isn't required behavior.
With Exchange, I admit my knowledge is weak, but I do believe that the default is to prefer remote storage over local storage.