Thank you for considering moving to GitLab.
Let me explain the terms first.
GitLab Free is the SaaS (Software as a Service) product hosted on GitLab.com while GitLab Core is a self managed (self-hosted/on-permise) variant.
SaaS is hosted by ourselves and totally free of continuous technical participation by your side. In case of SaaS (GitLab.com Free) the software is accessible via the internet. Also, you have no interaction with the server whatsoever and all the technical elements associated are handled by our team. Handling the SaaS product requires little or no technical knowledge as our team takes care of those elements.
Self-hosted software has to be downloaded and installed directly into your server and requires continuous and ongoing monitoring and maintenance from your side. It offers complete control which could be the reason choosing it over SaaS.
You can read more about differences between self-hosted (Core) and SaaS(Free) in our blog post explaining Why deploy on-permises in the SaaS era?
Also, there’s a full list of features included in each version (Core/Free and others).. Please check that out and see which versions works best for you.
I hope this answers your question, please let me if I could help you with other questions.