Neural networks are actually somewhere in between. They don't directly operate on symbolic expressions or explicit logical rules. And while they rely on probabilistic aspects for training (and sometimes for inference), they rely more on continuous-valued transformations in extremely high dimensional spaces. But if your goal is human-like intelligence, they are a pretty good bet, because we know the human brain also doesn't perform symbolic reasoning at its core and these things only emerge as high-level behaviour from a sufficiently complex system. But it also makes neural networks (and us too) prone to failure modes that you would not see in strictly symbolic reasoning processes.