One of the key features of human intelligence is being able to very quickly understand and solve problems on first impression, with no directly relevant experience or knowledge to pull from. Insight.
There's also the data representation problem. People can store, categorize, and process any kind of information.[1] The problem with both symbolic systems and "neural network"-type models is that they don't really allow for this. The problem with symbolic manipulation is more obvious: the categories of things and ideas are flexible, indeterminate, and innumerable, and a system that captures some subset of the complexity falls apart when you add a new, confounding, fact. Neural networks too are fragile, made-for-purpose, and fail to solve the above-mentioned first impression problem. You can look at a chair that looks nothing like any chair you've ever seen, yet realize that it is a chair.
[1]My own little theory is that consciousness is a crucial aspect of this. In the absence of a more specialized system in the brain (such as for language), any kind of information can be stored and remembered as a conscious experience. I don't pretend to be well-versed enough in the relevant neuroscience and philosophy to evaluate how plausible this is.