The breathings are important because they can change a word, kinda like when "h" changes "ear" to "hear".
The accents show how to pitch your voice. Acute: raise pitch (not volume); Grave: lower pitch; Circumflex: quick up and down pitch. Many people treat all accents the same in pronunciation, simply using them as stress marks. That's mostly what I and my children do.
I've heard that Greek was known as a "musical" language and that the accents were supposed to help non-native speakers "sing" it properly. But accents were not enough; no one today knows how to duplicate the classical Greek pronunciation. (Edison was born 2000 years too late.)
For beginners, the accent helps in memorizing the basic forms. For advanced learners, or those lucky enough to have a good teacher, the accents help with Greek verse (pomes).
I'm having a mild amount of trouble understanding the accents in Greek. I understand where they go and how they work, but what difference do they have on (a) pronunciation and (b) meaning, if any? The White book seems a little unclear on it - from what I can see it details the effect of rough versus smooth breathing, but it just seems to say where accents can go in a word and not what effect they have.