Mithilakshar — also called Maithili, Mithilaksara, and Tirahut — has been used since the 14th century in the northeast part of India, although it has limited use now. It is very similar to Bengali, so much so that early 20th century printers sometimes augmented set of Bengali metal type with newly cut Mithilakshar characters. It became distinct from Bengali by the 14th century.
Many of the glyphs have the same form in the two writing systems, but many do not. Some of the glyphs look the same, but are pronounced differently. (The Bengali “ba” and the Mithilakshar “ra” use the same glyph shape, for example.)
The two languages also have very different conjuncts — ligatures of two or more glyphs that most (all?) of the Brahmi-derived languages have in abundance. (Conjuncts are a way of both economizing on space and reducing the need for viramas to “kill” vowels when there are consonant clusters.) For example, the consonant cluster “ccha” exists in Mithilakshar as a single glyph (see the glyph at the top of this posting) with the two parent glyphs smooshed together vertically; in Bengali, they are smooshed together more horizontally.
Mithilakshar, like many of the northern Indian scripts, fell out of common use in the 20th century due to the rise of Devanagari.