Since the "official" things have already been addressed, to put up simply, Bitcoin Core (bitcoincore.org) is a site about the software/wallet, while bitcoin(bitcoin.org) is about the cryptocurrency. For Bitcoin Core related applications, you should use bitcoincore.org.
Also, if we take a look at the about page of each site, we can distinguish the differences, and know what it is all about.
Bitcoin Core is an open source project which maintains and releases Bitcoin client software called “Bitcoin Core”.
Bitcoin.org is dedicated to help Bitcoin to develop in a sustainable way.
Bitcoin.org is not Bitcoin's official website. Just like nobody owns the email technology, nobody owns the Bitcoin network. As such, nobody can speak with authority in the name of Bitcoin.
This differentiation was also addressed here in other thread by achow101:
Bitcoincore.org is Bitcoin Core's official website. As part of separating Bitcoin Core from Bitcoin.org (presumably as part of efforts to distinguish between Bitcoin Core the software and Bitcoin the cryptocurrency), the maintainers no longer post Bitcoin Core releases to Bitcoin.org. However the maintainers of Bitcoin.org can choose to mirror the binaries on their website and do so as they wish. You should direct people to bitcoincore.org as it is the project's actual website.