ANNs don't compress, they generalise.
Often this leads to compression, i.e. the internal generalised representation is smaller than the original input, but not necessarily. Imagine a ANN that is trained to use the screen input of a computer game to play it. If the game is very rich and conceptually deep the internal representation of a single screen input might be a lot bigger than the input itself, because ANNs put the single data points into the context of the overall data. Which leads us to the second point:
ANNs (and the neocortex) model data hierarchically.
This is what makes them so powerful. So it is not just about having a large number of parameters, they also have to be arranged in such a way that they capture the structure of the data (or the world), which very often seems to be hierarchical. Just look a two different pictures of a duck. On the pixel level they might be as different as random images, all the similarities emerge in higher levels of the hierarchy, when enough pixels combined give you the patterns of feathers, beak and webs. Bloom filters obviously lack this property. They would only give you a pixel by pixel account of whether you have seen (almost) exactly this picture before.