Difference Between Tensor and Tensor field?

The difference is all in your head. Literally.

The difference in calling the same object $A$ a "tensor over $\mathfrak{X}(M)$" as opposed to "a tensor field over $M$" is that the former emphasizes the fact that we have an algebraic object: a tensor over some module, while the latter emphasizes the fact that underlying the module there is some manifold and geometry is going on there.

Calling something a tensor field instead of a tensor forces you to remember that $\mathfrak{X}(M)$ is not just some arbitrary module, but that its elements can be identified with smooth sections of the tangent bundle of some manifold. These additional structures are occasionally useful.

Let's start with some simple explanations.

A tensor is "a mathematical object analogous to but more general than a vector, represented by an array of components that are functions of the coordinates of a space."

With each point in space , one can associate a set of scalars called a scalar field , and a bunch of vectors called a vector field , and one can also associate a set of tensors which would be called a tensor field .

A tensor field has to do with the notion of a tensor varying from point to point . A scalar is a tensor of order or rank zero , and a scalar field is a tensor field of order zero . A vector is a tensor of order or rank one , and a vector field is a tensor field of order one .

Some additional mathematical details.

$\mathbb {R}^n$ is a vector space representing the n-tuples of reals under component-wise addition and scalar multiplication .

A manifold is the natural extension of a surface to higher dimensions , and to spaces more general than $\mathbb {R}^n$ .

A tensor field on a given manifold M assigns to each point of M a tensor which is defined on the tangent space at the given point.

More precisely, a tensor field of type $\left(\begin{array}{c}r \\s \\\end{array}\right)$ on a manifold M is a mapping $T$ taking r differential fields and s vector fields on M to real-valued functions $f$ of class $C^k$ (having continuous partial derivatives of a certain order $k$ at each point ) on $\mathbb {R}^m$.

See also my related answer:

https://www.quora.com/In-laymans-terms-what-is-a-Tensor-Field/answer/Emad-Noujeim