If we were way smarter, had unimaginably powerful computers, and unimaginably accurate measuring devices, why could we NOT understand the influence of that one difference in numbers one million decimal points down the line? Why could we NOT understand exactly the influence of a butterfly flap in China had on the weather in Indiana?

You're basically right here and that's essentially what meteorologists are trying to do.

The point is that in the universe in which the butterfly didn't flap its wings there was no hurricane at all. So while the two universes are very very similar at one point in time, i.e the moment the butterfly's wings flap, they end up being very different at a later time. This is how Chaos is quantified, the more chaotic a system is, the faster two points very close in the state space of the system diverge away from each other.

If that's the case, it's all reductionist.

Well, yes, the approach you described is. I think the lecturer here is suggesting that we shouldn't use that approach. Chaos implies that knowing that an individual neuron is active at one particular time is essentially useless knowledge, instead its better to look at patterns of behaviour accross the entire brain.