There is no single correct way to count words. CJK text is normally counted by character and English by whitespace-delimited word, and whether punctuation, spaces and numerals count varies by publisher. Follow whatever definition your target platform gives; the several figures here exist so you can match any of them.
Text Diff is a literal comparison by line and by word; it does not understand meaning. Paraphrases and reordered clauses show up as large differences, while a critical change to a number or a unit may show as a single character. Important documents still need a human read.
Markdown to PDF renders with the fonts available in your browser, so output depends on your system fonts. When the typography has to match across devices, treat the generated PDF as the authority rather than the on-screen preview.