It is because WebMail 1.0 stored your outgoing email in the "OUTBOX" folder, not the "Sent" folder. You can find all your outgoing email in WebMail 2.0 after selecting the "OUTBOX" folder.

 Since nowadays most email client software no longer stores outgoing email in "OUTBOX" (or outbox, Outbox) folder, for consistency and convenience you are advised to move all email from the "OUTBOX" folder to the "Sent" folder.