Difference between spinor and vector field

How do we distinguish spinors and vector fields? I want to know it in terms of physics with mathematical argument.

Answer

The answer to your question depends on the context, but the basic unifying theme distinguishing different kinds of fields (like vector fields, scalar fields, etc.) is how these fields transform when they are acted on by Lie Groups (and or Lie Algebras) which falls under the mathematical subject of representation theory of Lie groups and Lie algebras. Here are some examples to illustrate what I mean:

Consider a real field $V^\mu(x)$ defined on 4-dimensional Minkowski space $\mathbb R^{3,1}$. We say that this field is a Lorentz 4-vector field provided when it is acted on by a Lorentz transformation $\Lambda = (\Lambda^{\mu}_{\phantom\mu\nu})$ (an element of the Lorentz Group $\mathrm{SO}(3,1)$), it transforms as follows: $$ V^\mu(x) \to V'^\mu(x) = \Lambda^{\mu}_{\phantom\mu\nu} V^\nu(\Lambda^{-1}x) $$

Now Consider instead a field $\Psi^a(x)$ defined on 4-dimensional Minkowski space $\mathbb R^{3,1}$. We say that such a field is a Dirac Spinor provided when it is acted on by Lorentz transformations, it transforms as follows: $$ \Psi^a(x) \to \Psi'^a(x) = R_{\mathrm{dir}}(\Lambda)^a_{\phantom a b}\Psi^b(\Lambda^{-1}x) $$ where $R_{\mathrm{dir}}$ is called the Dirac spinor representation of the Lorentz Group.

There are also other kinds of spinors, all of whom transform according to different representations of different Lie groups.