What is the difference between linear mappings and linear functions? Let $V$ and $V'$ be vector spaces over a field $K$. A linear mapping $$f:V \to V'$$ is a mapping which preserves addition and scalar multiplication.

My question is: what is the difference between linear mappings and linear functions?
Depends on the definition you are using. Linear mappings are typically the same as linear functions, although in some contexts, a linear function is strictly some function in the form of $y = mx+b$. 

Linear mappings are functions that map two domains and whose operations have linearity, that is the linearity of scalar multiplication and addition.
For the definition of linearity it still the same for functions or mappings.

But the term mapping has more general uses than than function, a mapping could be a function, functional or operator... 

However, these three terms are not the same.
 A function is a term you used for mappings that map a set A to B where A and B are subsets of real of complex numbers or almost any map into $R^n$ or $C^n$ .
 A functional is a mapping that is defined from a subset of a space X into the space of real or complex numbers.
 Finally an operator is defined from a space X to a space Y, where X, Y could sequence or function spaces.

Here an examples that clearfy more the difference between these three terms, 
let the function $f:R \rightarrow R $ such that  $f(x)= x$. 

Let us integral this function between $0$ and $1$,  then $ \int^{1}_{0}x =\frac{1}{2}$.

You can notice that the defined integral is a functional, it mapped a function ( x ) into a real ( 1/2).

However the undefined integral of this function is a function,  $\int x =\frac{1}{2}x^2$.

Thus, the undefined integral is an operator.

You can use the term of mapping for all previous examples but the term function can be uses only for the first case.