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A neural network is a several linear transormations $L_1,\ldots, L_m$ that are sequentialy appilied to feature vector $X$. A compositon of linear transformations is a linear transformation. So after all we get $L X$ where $L$ is a composition of $L_1,\ldots, L_m$.
The question is: if eventually we have that neural network is just applying a liner transformation to a feature vector what is the essential difference betwen neural networks and linear regression
No, a neural network is not several consecutive linear transformations. As you note, that would only result in another linear transformation in the end, so why do many instead of one? Actually, a neural network performs several (at least one, but possibly more, depending on the number of hidden layers) nonlinear (e.g. sigmoid) transformations.
That is also the difference between a neural network and a linear regression, since the latter uses a linear combination of regressors to approximate the regressand.
