I don’t know how to best explain this without confuse you further… I wanna say bokmål and nynorsk is closer than what Texas English and Scottish English is, but more different than what Australian English and British English is… probably. In short it’s two different attempts to write down how Norwegians speak. The Bokmål standard reflects on the continental influenced dialects spoken in costal Norway, while the Nynorsk standard reflects on the still-Norse-influenced dialects of inner Norway. Few or none actually speak bokmål or nynorsk, but the’ll have dialects fitting one written form better than the other, and use the grammar and lexicality from that one.
The differences aren’t all that big, at least not in modern time, so if you understand one you certainly understand the other. And as mentioned few actually speak any of them, Norway consists of a gazillion dialects and since all these are spoken in radio and TV you learn to understand them all (including Swedish and Danish) from an early age. I come from a county where the grammar is closer to Danish, and the pronunciation closer to Norse. My dialect fit neither bokmål nor nynorsk. I can’t really relate to the heated wars between those preferring one over the other. I see it more as a West VS East conflict, while us southeners wonder why they all speak so strange… ;) I was kinda intrigued to learn that Greece used to have the same situation, language wise. They had the “katharevousa”, the purified language where all foreign (read: Turkish) influences were removed and where the idea was to go back to ancient Greek. This is very similar to the idea behind Norwegian nynorsk, except here they tried to get rid of all Danish influence and went back to Norse. At the other end of the scale they have “dimotiki” (“folksy” Greek), where the ancient Greek has been mixed up with foreign influence, simplified modern grammar and other “errors”. This is closer to Norwegian bokmål. Dimotiki is what is used in Greece today, as the katharevousa was abolished in the late 1970s. Within poetry and within the church some still favour katharevousa, but it’s not taught to children and hence new generations won’t learn it. In Norway both spelling versions is taught to children, though the one closest to the dialect is what’s used for 90% of the teaching, and you’ll only have some lessons in the other. The majority of Norwegians - ca. 70% - use bokmål, while ca. 30% use nynorsk. If you want to work in the government you’ll have to master both, as you’re required by law to answer a letter/email in the standard used by the author of the letter/email.