Can someone please help me understand what the difference is between these two things? If I have purchased service cloud - would I see this in the list of APPS?
Service Cloud is your whole Salesforce instance / org.
Service Console is just one of the Apps in your Service Cloud. You can access Service console from Apps. It is a tab-based workspace that lets you manage multiple records on a single screen.
The first screenshot (1) is normal page layout. The 2nd one is your service console.