An Integration App offers much more than a Quickstart integration template. If you are planning to integrate two applications in a standard way (assuming you have an Integration App for those applications), then it is always best to go with the Integration App.
With an Integration App, the integration will receive bug fixes and new functionality and settings. It will have documentation and specific domain support – all the basics that come with any other fully managed application. In contrast, templates get none of that. A template will have a lot of the same initial integration functionality built out, but once a template is installed, you can do whatever you want with it.
All the managed components belonging to an Integration App (that are locked from being changed – to preserve the ability for the Integration App developer to upgrade them) are completely unlocked in a template.
If there are no Integration Apps available or if you are planning to use an application or integrate two applications in a nonstandard way, then a template can be a better choice.