Big-Omega and small-Omega question I found this on page 538 (resp. 576) of Introduction to Modern Cryptography 2nd (resp 3rd) edition by Katz and Lindell. The third bullet point under Example A.6 says:
$$\begin{array}{l}\text{Let }f(n)=n^4+3n+500.\text{ Then:}\\
\quad\text{(…)}\\
•\;\; f(n) = Ω(n^3 \log n).\text{ In fact, }f(n) = ω(n^3 \log n).\end{array}$$
I am just looking for a high level explanation, not a mathematical proof or something, about what this means.
The first line in the quote now in the question defines a function $f$ of variable $n$. The last line is about how fast (the output of) that function grows when $n$ grows towards $+\infty$.
Both statements $f(n) = Ω(n^3 \log n)$ and $f(n) = ω(n^3 \log n)$ are about how fast $f(n)$ grows relative the function $g$ defined by $g(n)=n^3\log n$. The first statement roughly tells that $f(n)$ grows no slower than $g(n)$, while the second tells that $f(n)$ grows faster. Both statement are true, and easily shown so when we look at the definition of $f$ and $g$.
Update following comment:

The statement is strictly about the value taken by $f(n)$, not about the difficulty or cost to compute $f$. What that value represents is not stated by the notation. It's state by what defines what $f$ represents. That could be time, or effort, or memory, or the value of an integer, like an RSA public modulus; we can't tell.
&quot;grows faster&quot; is not really the same as &quot;greater&quot;. Sorry, to be precise, we need the mathematical definition.

That's using a so-called (Bachmann-)Landau notation, which makes an unusual use of the $=$ sign: equality usually features transitivity. It takes some time to get used to it.
Anything more specific won't meet the &quot;high level explanation&quot; requirement in the question. But if we read a few lines above the question's quote in the book, or about (Bachmann-)Landau notation, the definitions are given.
It can be shown $f(n) = ω(g(n))\;\implies\;f(n) = Ω(g(n))$. That's why the sentence has $\text{in fact}$, often used to introduce a stronger or more precise statement, like: That's good. In fact, it's delicious.