Difference between a polynomial of degree $n$ and an $n$- tuple with the $n$th component $\neq 0$? When a polynomial $f := a_0 + a_1 X + \cdots + a_n X^n$ over a field with $a_n \neq 0$ cannot be regarded as a function, what is the major difference between $f$ and the $n$-tuple $(a_0,\dots, a_n)$? It seems that the plus sign and the indeterminate are redundant here.
 A: I mean, it's just a notational difference.
There are always multiple ways to notate any particular idea/object in mathematics, and you can even make up your own, so asking the difference between two different notations for the same object is kind of a weird question - they are different, so their difference is nonzero, and if you're comfortable with both, then that's never going to hurt you.
On the other hand, let me point out one advantage of using the classical notation $a_0 + a_1X + a_2X^2 + \cdots + a_nX^n$. The advantage of this is that it looks like a sum of "numbers", and sums of "numbers" is generally a "number". Since you can multiply "numbers", you should be able to multiply polynomials. Written in this classical form, it's easy to figure out how to multiply two polynomials - simply apply the distributive law multiple times. On the other hand, the tuple notation doesn't give any hint as to how one might "multiply" two polynomials. Do you multiply the corresponding entries in each tuple? What about multiplying two tuples of different lengths?
A: In fact none. The `formal' way of defining the polynomial ring over say a commutative ring $R$ is to identify it as the collection of all set functions $f:\mathbb{N}=\{0,1,2,\ldots\}\to R$ such that $f(n)=0$ for all large $n$. If $f,g$ are two such, we can define $(f+g)(n)=f(n)+g(n)$, $(fg)(n)=\sum_{i=0}^n f(i)g(n-i)$ and degree of $f$ is just the largest $n$ such that $f(n)\neq 0$.
