Neural Networks are models with adjustable parameters that can be trained using optimisation techniques like backpropagation alg. to solve classification and regression problems whereas GA are stochastic optimisation techniques to solve optimisation problems.
As Mohammed said, AG and ANN there are two different group method of AI difficult to compare. If you looking for good library of ANN my recomendation is FANN (http://leenissen.dk/fann/wp/), as far as I remeber for fuzzy ANN too.
If are you looking place to implement fuzzy in AG that selection method will be good option.
GAs has to be used for minimization and maximization for global search and also for
getting the feasible solution in CSP while ANN has to be used for the classification to achieve the solution. So we can use the GAs, AGA, SGA, GA with hill climbing for the training the NN.
As other scholars said, GA, ANN and FIS (either Mamdani or Sogeno type) are in different categories of Artificial Intelligence (AI) tools. GA is useful to optimize a problem when you have a KNOWN fitness function (a formula, a game, a simulation, etc.). However, ANN is usually used to model, predict or guess relationship between different data (e.g. you can use it for forecasting purposes). ANN can not be used for optimization in most of the cases.
In other category, through FIS scholars tried to learn a computer "human-like reasoning style". Computers understand crisp numbers, for example in your mind, 5.9999 may be very close to 6.00000 but for a computer (without FIS) 5.9999 are different from 6.0000 as the 5.00001 and 6.0000 are different. It means any number less than 6.0000 is not acceptable (in this example).
For the last section of your question, GAs are widely used to select some parameters needed to design ANN. For example you need to determine the number of layers or the number of intra-nodes of the ANN. Some practitioners used their own expertise or some simple try and error methods to guess these parameters. However, you may use a GA to calculate a hopefully, near optimal set of parameters for the ANN.
FIS is merged into ANN, as well and adaptive neuro-fuzzy inference system (ANFIS) is the result. ANFIS is an estimator tool consist of some Sugeno type fuzzy If-Then rules learning nonlinear relationships using an ANN. The If-Then rules which are the most confusing part of designing an FIS, are drived using an ANN training system.
Thank you all for your suggetion and specilly to Seyed Mahdi Homayouni because It clear lot of my doubts. It's really a nice explanation. Can you give any suggestion when it comes to genetic fuzzy system specially for Mamdani type fuzzy system?
To complement the others answers. It is worth noting that GA search a space of all possible solutions. Each individual in the population represents a solution in the search space and hence a certain parallelism is introduced.
Neural network are modelled on the human brain and how the cells interact with each other. It represents a graph with an input and an output.
Interestingly, Cartesian Genetic Programming has recently evolved Neural Network successfully. These outputs have been optimised and often do not have any hidden part.
Although EAs and Neural Network are very different optimisation techniques, they could too work well together to solve problems. Perhaps this type of learning phase could help you with your fuzzy inference System.
GA and ANN are two different tools. ANN learns while GA evolve. See ANN Back Propagation to get a feel of ANN. And see Hollands Book to get a feel of GA evolution idea.