Each of your email clients keep independent copies of mail downloaded from a server. Because messages can be deleted on a server by one client, other clients may not get copies of some messages.
All of your email clients use one mailbox -- the one on the server. If a message is deleted or moved on one client, other clients will see the same action, though there may be a time delay. This means that no matter how many clients you use, all of the email clients see the same view of the mailbox. This is good for smartphone users because the phone, desktop at home, and webmail from the office all have the same view of the mailbox.