I'll try my best. I didn't design the brains
Assembly was Andrej's latest experimental version. It was intended to replace the MLPBrain. It includes some experimental operators, but is unfinished, and thus far I haven't seen a need for it. I think it was supposed to be faster than the MLP
DWRAON, stands for Damped, Weighted, Recurrent, AND-OR Network. It was the one we started with, and is described in his video of Version 4 above. Each box has several CONNectionS, and chose either to be an AND type or an OR type, thus multiplying or adding the different, weighted connection values together. It had some limitations, such as flooring or exploding connections, making the bots behave "jitterly".
MLPBrain, stands for Multi-Layer Perception. Not sure where the name comes from, but it's basically the DWRAON but without Or connections, and every brain box is sent through a mathematical function called a sigmoid, which essentially allows us to feed every brain box very large positive or negative values to get a wider range of responses. It's very natural (no brain connection is ever truely "off" or "on" [0,1]) and produces more responses to changes in the environment. It is the current brain used by both me and Andrej's latest versions.
Hope that answers your question!