Connecting live to Tableau Server data sources means that you are using the same, and most recent (updated), data as everyone else. With this in mind Tableau Server is used to make data centralised and easier to manage, where as if you create an extract you are taking a snapshot and thereby making your own local copy. In other words if anyone makes changes to the live connection on Server yours will be outdated and not be the same version of the "truth".
However during dashboard development we tend to create extracts and work offline, this avoids you having the query data over a slow internet connection and make the report creation quicker. Once the dashboards are done you can easily replace this connection with the live one.
Hope this helps.
To be honest, the same rule of other databases connections (live or extract) is applicable to Tableau data server also.
I believe based upon the no.of developers using the published data source & size of data source in Tableau server - Extract & Refresh schedule is managed.