While both Live Stream and Notifications will show you recent engagements across social channels, the Live Stream shows you the most recent engagements with all the context that you need to react or respond to them.
For every social activity, Live Stream will show you the number of times a particular Connection has engaged with your Brand. You can view the person's followers and the number of people he or she is following.
With CRM integration, you can find out if the person is a CRM Lead or Contact. Also, you will see available internal actions such as Follow (for Twitter and Instagram). You can +Add to CRM, and Discuss, to talk about the interaction with your team collaborators.
If you click on the Know more link, you will be able to view more details, such as their recent interactions with your Brand, and their recent posts on the channels you have associated with them.
To directly engage with a post, hit Reply, Like, Tweet, Retweet, or Direct Message. On the other hand, social notifications will only display the social actions of the audience such as likes, tweets, comments, and direct messages.