A shared mailbox acts a standalone mailbox. Emails are received into it. The shared mailbox can have its own set of folders and calendar entries. Multiple people can have access to the mailbox and multiple people can send from the mailbox. This will show as a separate mailbox.
Distribution lists allow emails to be received into one single email address and then are distributed amongst a number of other email addresses. Each person in the distribution list will receive their own version of the email.