Thank you for the question and I apologize for the delay in getting back to you.
There are two main differences between J!WHMCS Integrator and Integrator 3:
Expandable: For most customers, it isn't necessary to have more than Joomla and WHMCS integrated together, so for them J!WHMCS Integrator is ideal. For some customers that use many applications or plan to do so, Integrator 3 is the right choice. Integrator 3 permits you to expand to integrate multiple Joomla / Wordpress sites with the same WHMCS or Kayako Fusion site.
Page Mapping: I know, what is that... page mapping is the technique of mapping a specific menu item in Joomla or Wordpress to a specific page in WHMCS. J!WHMCS Integrator only maps to essentially one page in Joomla, so any content you want to display in WHMCS must be displayed on all WHMCS pages. With Integrator 3 it is possible to specify a menu item in Joomla or Wordpress to apply only to a certain page in WHMCS. This allows you to include customized modules and content for use on the pages you want in WHMCS, best of all this content is controlled in Joomla or WHMCS, not hard coded in WHMCS tpl files.
Those are the main differences but there are other minor differences as well.
As far as which templates to utilize, you can use any template providers, though some have had issues using the Gantry framework on some of the templates provided by RT. They are great templates, but some of the styles will need to be undone when wrapped around WHMCS (for example, the radio buttons sometimes get hidden and need to be un-hidden if that makes sense using css).
Let us know if we can be of any further assistance!