Why polynomial interpolation is considered as better than others?
In case of interpolation, the function $\phi(x)$ to approximate the unknown function $f(x)$ may be polynomial, exponential, trigonometric sum, Taylor's series, piece-wise polynomial etc. Then why polynomial interpolation is considered as better than others (although I know that there is a justification for approximation by polynomials (Weierstrass's theorem), but it does not true that there is no justification to approximate by exponential, trigonometric etc).
Some where I read, reason for considering the polynomials in the approximation of functions $f(x)$ is that the derivative and indefinite integral of a polynomial are easy to determine and are also polynomials. But if $\phi(x)$ is exponential, trigonometric sum, Taylor's series, piece-wise polynomial etc then also derivative and indefinite integral are determined like polynomials.
Then what are other reason(s) for taking polynomials into consideration?
The other reason is probably that comparing the polynomials to other elementary functions the polynomial are also easier to compute, this is a practical justification of course. Theoretically speaking... i'm not sure of this statement because you can also approximate a function using the trigonometric polynomial. Trigonometric functions aren't easy to compute in practice (in a computer i mean) but of course are important for solve in closed form certain differential equations. I don't know about the exponential, but they have other applications, like in functional analysis. I believe it actually depends from the application rather than only theoretical properties.
I don't know your background of course but if you try to implement from nothing a trigonometric function (in its whole domain with enough accuracy) you would see a lot of issues...