All four styles agree on most of a headline. They capitalize the first and last word, and every noun, verb, adjective, adverb and pronoun, short verbs like Is and Be included. They lowercase the articles a, an and the. The arguments are about the small words that join the rest together.
Prepositions: length or job?
AP and APA go by length: prepositions of three letters or fewer (of, in, on, to, by) stay lowercase and longer ones are capitalized, so With, From and About. Chicago and MLA go by job: a preposition stays lowercase however long it is, so with, between and without.
Conjunctions and the word to
Everyone lowercases and, but, or, nor and for. AP and APA also lowercase so, yet, if and as, being three letters or fewer. MLA lowercases so and yet; Chicago capitalizes them, but keeps to and as lowercase in every use.
Colons, hyphens and phrasal verbs
The first word after a colon starts a subtitle and is capitalized. Hyphenated words get a capital after the hyphen (Self-Driving), except that Chicago keeps the part after a prefix lowercase (Re-election). A short word in a phrasal verb, as in Set Up or Log In, is an adverb, so every style capitalizes it; the converter knows the common ones.
Sentence case capitalizes only the first word and names; it is what AP uses for its own news headlines. The full comparison is on the capitalization rules page.