What’s the difference between Classifier and Hopfield Network?

A Hopfield network classifies a N \times N of matrix data to associated matrix \{ A_1, A_2, A_3, \dots, A_n \} of order 2^N - 2. It is made up of inter-connected nural nets with with no input/output.

A classifier takes an input vector \vec I = \left [ \begin{matrix} a_1 \\ a_2 \\ \dots \\ a_n \end{matrix} \right]. With mean sample spaces \vec \mu_- \stackrel{\text{def}}{=} \frac 1 m_- \sum \limits_{y_i=-1}\vec x_i and similarly \vec \mu_+ \stackrel{def}{=} \frac 1 m_+ \sum \limits_{y_i = +1} \vec x_i

Using the sign of f(x) to classify them as a_- \,or \,a_+

f(x) \stackrel{def}{=} || \vec \mu_+ - \vec x ||^2 - ||\vec\mu_- - \vec x||^2 = 2 \langle {\vec \mu_- - \vec\mu_+ , \vec x}\rangle + || \vec \mu_- ||^2 - ||\vec \mu_+||^2

Clearly, it’s a vector, it can be easily extended to classify more of it a_n.

But, what it makes them different? Both Hopfield and Classifier is doing same work, so what is the point of using them both?