The vowels in 'book' and 'good' are both represented by an inverted omega in the international phonetic alphabet (ʊ), according to the OED. Can it not express their difference? 'Isaac Pitman's Shorthand Instructor' uses the sentence 'That book is not much good' to show the six short vowels in English. Maybe he presupposes a different pronunciation of 'book', but how is that represented in the IPA? The symbols used by the Pitman book are ĕ (book) and ŏŏ (good).
Some people pronounce book and good the same, so there is no reason to differentiate them. Other pronunciations vary minorly, and can be created in International Phonetic Alphabet by usage of diacritics. Some people pronounce book like ʊ̜, which is the inverted omega you mentioned earlier plus a less rounded/unrounded diacritic (probably doesn't show up well). That sound is fairly rare. So, yes, IPA can express the difference in accents where it exists, but not well. https://en.wikipedia.org/wiki/Near-close_near-back_unrounded_vowel gives you other possible characters for the sound.