There is no difference in the two; earlier it was 3dvia Composer and now it is under SolidWorks product flagship so it is SolidWorks Composer
Well so far according to my VAR a 3DVia version will not pull a license from a Solidworks Network License manager. You have to install the SW verision. So off I go again. 2 months and cannot get the Network License Versions running. Its either the Network Manager or the Version. I dont think they really know either.
It can be a little confusing at first but essentially the move to SolidWorks composer was to avoid such frustrations as having to run 2 license servers or having to fill in a form to transfer a standalone license.
To run earlier versions of a 3dvia Composer Network License you need to run a seperate license server.
To run SolidWorks Composer Network license you just add the serial number (presuming it has been updated) to your SolidWorks Network License Manager.
This is done via the License Administration tab-> Modify.
There are still a few potential hiccups on the client side but these are rare and your reseller should be able to find solutions via the knowledgebase as there is a checklist of things to solve any issues in the transition.
Also standalone licenses are much simpler for SolidWorks composer, just like solidworks you can transfer licenses without filling in any forms etc that you had to with 3dvia Composer.
Another reason for the change was to align the releases of SolidWorks with the releases of Composer. Now a version of Composer that supports the latest version of SolidWorks files is released at the time. Previously there was a delay of 2-3months.
To add to Alan's comment, pulling a license for Composer using the SolidNetwork License Manager, the SNL Manager must be running 2013 SP2 or newer and your SolidWorks Composer must also be 2013 SP2 or newer.
If you are using the older 3DVIA Composer, it will not be able to pull licenses from the SNL manager and requires it's own FlexLM license server.