Why to choose to work with a functional instead of a function? Why to choose to work with a functional instead of a function?
Notice in a function you evaluate points and in functional you evaluate functions.
and why a linear functional is important in general ?
A functional $\phi(f)$ is linear if the domain of its existence together with the functions $f(x)$ and $\psi(x)$ contain the function $af(x)+b\psi(x)$ and if the equality  $\phi(af+b\psi)=a\phi(x)+b\phi(\psi)$, and $a,b\in\mathbb R$ holds.

You pose a dichotomy that is not so. One does not look at functionals as opposed to functions. 


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*First of all, a functional is a function. 

*Second, you say that "functions evaluate points". That's not a good point of view. Typical calculus functions evaluate on numbers, as they usually evaluate as formulas. But it's very easy, even at the calculus level, to think of functions that evaluate on functions. For example, given a curve $r:t\longmapsto (x(t),y(x))$, $t\in[0,1]$, its length is the function $L(r)=\int_0^1\sqrt{x'(t)^2+y'(t)^2}\,dt$. So $L$ is a function that evaluates on functions. 
Linear functionals are important because it is extremely common in mathematics for certain objects to form a vector space. And it's been noticed that to understand a vector space, in particular when they are infinite-dimensional, understanding of its dual is important, and often times helps prove results about the space. That's the core of Functional Analysis, and understanding of topological vectors spaces (Banach Spaces, Hilbert Spaces, Frechet Spaces, Bounded Linear Operators on a Banach Space, etc., etc., etc. ) comes together with understanding their duals.
