Meaning of polynomially larger or smaller in the context of the master method
(This answer refers to the version 6 of the question, which does not contain &quot;my professor's response&quot;.)
It looks like there is some typo/inconsistency/misunderstanding somewhere in your class, the slides or your post. Hence, I will start from scratch, addressing the problems I can recognize.
Let me fix the meaning of &quot;polynomially smaller&quot; and &quot;polynomially larger&quot; in the current context. There are other usages, which are not relevant here.
Meaning of polynomial differences.
$f(n)$ is polynomially smaller than $g(n)$ if $f(n) = O(g(n)/n^\epsilon)$ for some $\epsilon &gt; 0$.
$f(n)$ is polynomially larger than $g(n)$ if $f(n) = \Omega(g(n)n^\epsilon)$ for some $\epsilon &gt; 0$. We will never say $f(n)$ is polynomially equal to $g(n)$.
Here are some examples.

$f(n)=1$ and $g(n)=n^2$. Then $f(n)$ is polynomially smaller than $g(n)$. This is what you believed and it is correct.
$f(n)=g(n)=n^2$. Then $f(n)$ is not polynomially smaller nor polynomially larger than $g(n)$.
$f(n)=n^{1+\frac{1000}{\log n}}$ and $g(n)=n^2$. Then $f(n)$ is polynomially smaller than $g(n)$.
$f(n)=\dfrac{n}{\log n}$ and $g(n)=n$. Then $f(n)$ is not polynomially smaller nor polynomially larger than $g(n)$.

$T(n) = 4T(\dfrac n2) + 1$
$a=4$, $b=2$, $n^{\log_24}=n^2$ is polynomial larger than the constant function 1. This is case 1 of the master's theorem.
$T(n) = 4T(\dfrac n2) + \dfrac{n^2}{\log n}$
$a=4$, $b=2$, $\dfrac{n^2}{\log n}$ is not polynomially smaller nor polynomially larger than $n^2$. None of the three cases of master's theorem can be applied. However, the extension of case 2, case 2b can be applied.

Simple exercises
Exercise 1. Suppose $f(n)$ is polynomial smaller than $g(n)$. Show that $f(n)$ is eventually smaller than $g(n)$, i.e., there exists some constant $n_0$ such that $\lvert f(n)\rvert \lt g(n)$ for all $n&gt;n_0$.
Exercise 2. Verify that $f(n)$ is polynomial smaller than $g(n)$ if and only if $g(n)$ is polynomial larger than $f(n)$, assuming that both $f$ and $g$ are always positive. 
Exercise 3. Suppose $f(n)$ is polynomial smaller than $g(n)$. Find another function $h(n)$ such that $f(n)$ is polynomial smaller than $h(n)$ and $h(n)$ is polynomial smaller than $g(n)$.