What is the difference between the LTS version and the stable version of node.js ?
What version should we use on production ?
It depends where you are using node.js in your apps and how sensitive are they.
LTS (Long Term Support, currently v6.11.2) - as was mentioned in the comment, will have support and maintenance for at least 18months. So its better to use this one in Production for node.js as a back-end service.
Stable (currently v8.3.0) - will have support for approximately 8months, with features/updates released more often. This version will be ok for Production if you're using node.js for fron-end services (dependency management etc.). Will work for node.js on back-end as well, if you have ability to easily update your apps without interrupting the environment.
The reason of 2 type node.js versions:
As node.js team described when they announced this feature is that 2 different types of node.js versions will meet your node.js needs. Basically, if you have a complex node.js app and you want stability, then stay on LTS. If you are able to update your app often, or you use node.js only for the fron-end tools, then use Stable. The philosophy is to push new features, performance updates, bug fixes etc. in a quick manner, so companies/developers who wants the 'on the edge' technology and loves node.js, will be able easily use it and expect updates every 2weeks.