Expert Systems are very different systems from Neural Networks
These systems differ in many ways, both with regard to their architectures and to their uses.
Expert Systems (at least in the traditional understanding of the word) are driven by [typically] high-level rules which the engine uses, along some input, to infer some conclusions about the input. Rules are typically entered explicitly, essentially translating some statements made by experts in a particular domain, into whatever predicate representation is implemented in the Expert System.
Neural Networks (NN), on the other hand, may have their topology manually/explicitly set, but otherwise are typically able to learn, automatically, how to associate some inputs or input sequences with particular outputs.
The above descriptions are certainly reductive of both concepts, maybe particularly of Neural Nets which come in many different forms and flavors (with many different usages, well beyond pattern recognition), but, I hope, will help you understand how very different these systems are.
One salient difference may be with regards to "transparency", whereby Expert Systems can typically output some "explanation" for their deductions ("Input 1 and Rule #3 indicate conclusion A, with a probabilty of 81%"), while NNs tends to be black boxes. This is not to say that NNs cannot come to a very precise "understanding" of their world (and such understanding gets encapsulated in the topology of the network and the relative weights associated with various inputs and/or neurons), it's just that this expertise about their world is not so readily translated into plain English. With regards to transparency, and
in very broad terms, an NN compares a bit to a mathematical formula, whereby an Expert System is more like a recipe book.
Another notable difference, as hinted above, is that an expert system requires the translation and loading of explicit rules ("If the temperature exceeds 200 degrees, the safety valve starts opening"), whereby the Neural Network discovers such "rules" based on the training sets supplied to it (such as labeled input sets).