Brahmi is sort of the Phoenician of East Asia: almost all the non-logographic scripts in East Asia come from Brahmi, including almost all of the scripts used in India, Tibet, Nepal, Mongolia, Thailand, Indonesia, Cambodia, and Laos. Brahmi was a big deal.
Like Kharosthi, Brahmi is an abugida. Either Kharosthi or Brami was the first abugida in the world, but it isn’t clear which was first — it’s difficult to date either of them well. There were many examples of Brahmi in around 250 BC from the stone pillars of the Edicts of Ashoka, and there was some geographical variation already, but not much. There are some examples of what might have been Brahmi on some pottery fragments from 500 BC, but not many and it isn’t completely clear that they are Brahmi.
The glyphs in Brahmi and Kharosthi are very different, but the mechanics are very similar: consonants are assumed to have the “a” sound attached to them; if it is a different vowel, a diacritic is added to the consonant. However, leading vowels are treated differently. In Kharosthi, there is a sign meaning “vowel at the beginning of a word” which is presumed to be an “a” unless it has vowel diacritics on it. In Brahmi, there are five different glyphs for five different leading vowels.
Brahmi also deals with consonant clusters in a slightly different way than Kharosthi: both smoosh the consonant signs together, but Kharosthi smooshes horizontally, while Brahmi puts one of the consonants under the other.
Brahmi might have descended from Phoenician via Aramaic. Many of the glyphs look similar to the Aramaic glyphs of similar sounds, although sometimes flipped (which is common when the writing direction changes, and Brahmi is a left-to-right language) and sometimes upside-down. To my eye, the upside-down-ness makes it easier to decorate those glyphs with the vowel diacritics.
While it is clear that Kharosthi developed in Pakistan, which definitely used Aramaic as part of the Achaemenid Empire, it’s hard to tell where Brahmi developed: the definitive records burst onto the scene with Ashoka (whose dynasty originated in eastern India), while the disputed texts come from southern India — neither of which were under Achaemenid rule. To my eye, the Brahmi glyphs look more like Aramaic than Kharosthi, despite Kharosthi coming from a Aramaic-writing area.