Live Session: it's a reserved area in the AppMon Server's Memory. The more memory you assign to the AppMon Server the more PurePath can be held in the Live Session.
Continuous Session: When the feature enabled, more PurePath will be stored in File System. Maybe couple of days or weeks. It depends on "Storage Quota".
You can store your session to be offline session, then you can imported to client without server.
It could be useful to repeat the a bug.
There are more features need to be mentioned.
For the 5.x version:
In server.config.xml under <settings>
optimizeforthroughput="false" in dT client's Storage page "Optimize for user triggered transaction analysis"
optimizeforthroughput="true" in dT client's Storage page "Optimize for dynaTrace Server throughput"
For the 6.0 version:
By default in 6.0 we use optimizeforthroughput="true" which means that as soon as a PurePath is completed it will be stored to disk (session store) and removed from memory.