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I know that polynomials can be refactored in terms of their roots. However, this must imply that two different polynomials have different roots (this is just what I think). So my question is: Are polynomials with the same roots identical? - if so, why?

No, they are not.

For instance, $2x^2-2$ and $x^2-1$ have the same roots, yet they are not identical.

And, depending on what you mean by "the same roots", we have that $x^2-2x+1$ and $x-1$ have the same roots, yet they are not identical.

Again, depending on what you mean by "the same roots", $x^3+x$ and $x^3+2x$ both only have one real root, yet they are not the same.

However, if two monic polynomials have the same roots, with the same multiplicities, over some algebraicaly closed field (like the complex numbers $Bbb C$) then yes, they are identical.

No, they aren't:

$f_1(x)=(x+1)(x-2)$ and $f_2(x)=5(x+1)(x-2)$ have the same roots. So if they are of the same degree, they must be identical up to a constant. But they don't even need to have same degree to have the same roots: $f_3(x)=x^2$ has the same root as $f_4(x)=x$.

For polynomials over $mathbbR$, the answer is no; for example, $f(x)=x$ and $g(x) = x(x^2+1)$ have the same roots over $mathbbR$—with the same multiplicities—but they are not equal.

For polynomials over $mathbbC$, the answer is almost. The fundamental theorem of algebra says that every polynomial over $mathbbC$ of degree $n ge 1$ splits uniquely into $n$ linear factors. So if $f$ and $g$ have the same roots $alpha_1,alpha_2,dots,alpha_n$, listed with multiplicity, then
$$ f(x) = lambda (x-alpha_1)cdots(x-alpha_n) text and g(x) = mu(x-alpha_1)cdots(x-alpha_n) $$
for some $0 ne lambda,mu in mathbbC$. So roots (with multiplicity) determine polynomials over $mathbbC$ up to a multiplicative constant and, in particular, monic polynomials over $mathbbC$ are uniquely determined by their roots.

For polynomials over finite fields, the answer is very much no. There are polynomials that don't just have the same roots, but they have all the same values for every input. For example, the polynomials $f(x) = x$ and $g(x)=x^3$ over $mathbbF_2$ satisfy $f(x)=g(x)$ for all $x in mathbbF_2$, and yet $f ne g$.

The multiplicity counts too: for example $x$ and $x^2$ have the same roots, but are different polynomials. If two polynomials have all the same roots and all the same multiplicities, then yes they are equal. However, this statement needs to be interpreted correctly: you need to work over $mathbbC$ (or some other algebraically closed field). For example, over $mathbbR$, the polynomials $x^2+1$ and$ (x^2+1)^2$ have the same real roots (namely, they have no roots!) but are clearly not the same.

So: you have to count the roots with multiplicity in the algebraic closure.