Record, register, list, enroll, catalog can mean to commit to writing for the sake of immediate or future use.
Record usually implies as its purpose the making of an exact or official entry or statement which gives evidence of the facts involved; the act serves as an aid or a check to memory or as a means of supplying details unlikely to be remembered indefinitely.
Register usually implies accurate entry into a formal record of facts or particulars of a certain kind which require or deserve recording.
Sometimes register carries a further or a slightly different implication; thus, one registers a letter by payment of a special fee and obtaining a receipt to ensure its safe delivery by requiring a record of each person who handles it; a thermometer registers the temperature when the mercury expands or contracts to a certain mark in degrees.
List implies an entering in a list (as of names, figures, needs, or events).
Enroll may add to list the notion of setting apart those entered in a distinctive category (as members of a body or adherents of a person or cause) and therefore may connote a winning over, an enlisting, or an admission to membership.
Enroll is sometimes also used of things in the sense of record, but in its more usual extended sense, it also includes the connotation of honoring.
Catalog commonly implies an enumeration of all items making up a class or group, usually with descriptive or defining details.
Catalog may also imply assignment of an item to its proper place in a list or sometimes in a category.