Question: I am wondering about the differences. Based on my understanding, MLP is one kind of neural networks, where the activation function is sigmoid, and error term is cross-entropy(logistics) error. Looking for help, thanks!

Answer: 
You are right, MLP is one kind of neural network.

**There are several kinds of NN**, you can have a NN based on Radial Basis Function with a Soft gating strategy, for example. You can use a committee machine strategy to form a NN...
Multi-Layer Perceptron is a model of neural networks (NN). There are several other models including recurrent NN and radial basis networks. For an introduction to different models and to get a sense of how they are different, check this [link](https://en.m.wikipedia.org/wiki/Types_of_artificial_neural_networks) out.