ISO 20022 is the international standard for financial messaging, covered in full in the entry on ISO 20022 formatted messages. People use the bare name "ISO 20022" to mean the standard as a whole: its common data dictionary, its modelling method and the catalogue of message definitions built from them.
The standard is published by the International Organization for Standardization and maintained through a registration authority, with SWIFT acting in that role. Messages are grouped into business areas identified by four-letter prefixes, for example payments initiation (pain.001 and related messages), payments clearing and settlement (pacs.008 and others) and cash management (camt). In the SWIFT world these XML messages are often called MX messages, to distinguish them from the older MT messages they are replacing.
The difference between the two entry names is only one of emphasis: "ISO 20022" usually refers to the standard and the migration programmes built around it, while "ISO 20022 formatted messages" refers to the actual messages that banks and payment systems exchange.