Soldato
Shared Mailboxes are good if you have say 3 people in an accounts team and you want all 3 to receive emails to [email protected] and be able to send emails from that same address as well. Each shared mailbox can store 50GB of emails before requiring licencing. Shared mailboxes are displayed separately in outlook to your standard inbox.
Distributed mailboxes are good if you just want a group of people to receive emails to a single email address, so for the [email protected] above, each member would receive the same email directly to their individual inboxes but would not be able to send emails as [email protected] like they can if its a shared mailbox.
You can send as (or on behalf of) a distribution list you just need to assign the permission, you don't get it by default.