If you only use web-based access to your mail, you should see exactly the same screen wherever you log on from. However if you access mail using a 'POP3' mail client (such as Outlook Express, Outlook, Windows Mail, Windows Live Mail or Thunderbird) then the default setting is for it to download mail from your server, deleting the mail from the server as it does so. (i.e. you'll then be looking at the mail on your own PC, not on a remote server). So when you later examine mail on the server using web-based access, it will no longer be there.
If that's what's happening in your case (and you don't want it to) you'll need to configure your email client to leave a copy of your mail on the server when it downloads. Since you've not stated which program you're using, it's rather difficult to provide instructions. However the following works for Outlook Express. (Most other programs are similar):
Click on Tools, Accounts, Mail.
Click on the name of your email account, to highlight it.
Click Properties, Advanced.
Place a tick alongside 'Leave a copy of messages on server'.
Click Apply, OK.
Chris