For example. Given function
$f(x)=\sqrt[3]{x} \Rightarrow f'(x) = \frac{dy}{dx}= \frac{1}{3}x^{-2/3}$
$dy = (\frac{1}{3}x^{-2/3})dx$
compared to...
$\Delta y = \frac{1}{3}x^{-2/3}\Delta x$
I know its a silly questions but i am trying to have a consistent understanding of math symbology
If $\Delta x$ is a small interval (think of a small "change" in $x$), then it produces a corresponding change $\Delta y$, which by definition must be given by the formula $$ y + \Delta y = f(x + \Delta x). $$ In other words, since $y = f(x)$, this is equivalent to $$ \Delta y = f(x + \Delta x) - f(x). \label{diff} \tag{1} $$ Assuming that $\Delta x \neq 0$, we can rewrite this as $$ \Delta y = \frac{f(x + \Delta x) - f(x)}{\Delta x} \, \Delta x. \label{diffquot} \tag{2} $$ Notice that this is still an equation based on a secant line approximating the graph of the function $f$ between the points $(x, y)$ and $(x + \Delta x, y + \Delta y)$.
When we allow $\Delta x \to 0$, the second of these two points collides with the first, so we are no longer considering a secant line connecting two points on the graph of $f$. Equation $(\ref{diff})$ makes this explicit by showing that $\Delta y \to 0$ as well.
Surprisingly though, for many nice functions, Equation $(\ref{diffquot})$ is still useful to approximate $\Delta y$ as long as these quantities are small.
When we let $\Delta x \to 0$, so $\Delta y \to 0$ as well, we change notation to the differentials $dx$ and $dy$, so we have $$ dy = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} \, dx = f'(x) \, dx, \label{differential} \tag{3} $$ which is only a statement about predicting a change $dy$ in the variable $y$ along the tangent line for a change $dx$ in $x$. But if $\Delta x$ is small, this is approximately equal to the actual change $\Delta y$, hence $$ \Delta y \approx f'(x) \, \Delta x. $$
To answer your particular question, where $f(x) = x^{1/3}$ and $x \neq 0$, then $$ dy = \tfrac13 x^{-2/3} \, dx $$ and $$ \Delta y \approx \tfrac13 x^{-2/3} \, \Delta x. $$
Here you can play around with the what these various quantities look like interactively. (Instead of $x$ and $\Delta x$, they are called $a$ and $h$ in the graphic.)