Converse Inbox(Available with Grow) shows a list of all SMS Conversations and prompts/alerts on a new message, in a single list inside Salesforce very much like an email Inbox. You can configure the Inbox on your Utility bar in lightning or Classic side Panel in Classic version. The Inbox:-

• Alerts you of new incoming SMS
• Shows list of all (My) Unread and New Conversations 

Notification (Available with Start) – This show a list of all unread incoming messages and gives you the capability to reply to incomings. You can configure this component  on your utility bar in lightning or Classic version. The Notification:-

• Alerts you of new incoming SMS
• Shows list of all (My) Unread and New messages

It is a compact version of the Converse Desk with minimal functionality focusing on reading and replying to the unread and new messages. For Salesforce Lightning,  you can configure the Converse Inbox in the Lightning Utility Bar, and for Salesforce Classic, you can set it up in the Classic Side Panel. For more information about setting up Converse Inbox, see (Admin Guide Topic link).