In a simple sense, NTAG cards can do NFC things, but MIFARE can do lots more (access control for example)..and also NFC things..somewhat.
Magic mifare refers to special cards that let you bypass the write-lock of genuine mifare cards. These are mostly used for cloning keys (either for red-team pentesting or for people who want a copy of an office key for whatever reason)
MIFARE Classic uses a proprietary and mandatory encryption/authentication algorithm and is therefore not ISO 14443-4 compliant. As a result, NFC-compliant readers don't need to support it, and in fact non-NXP ones (including many popular Android phones) usually don't.
On the other hand, as you say, MIFARE Classic supports capabilities beyond NFC/NDEF, but there are fully NFC-compliant tags that do so as well (e.g. MIFARE DESfire, which properly stacks encryption in an ISO 14443-4 compliant way).