Newton: In this notation, due to Newton, the primary objects are functions, such as $f(x)=x^2\text{,}$ and derivatives are written with a prime, as in $f'(x)=2x\text{.}$
Leibniz: In this notation, due to Leibniz, the primary objects are relationships, such as $y=x^2\text{,}$ and derivatives are written as a ratio, as in $\frac{dy}{dx}=2x\text{.}$
Both notations are in common usage, and both notations work fine for functions of a single variable. However, Leibniz notation is better suited to situations involving many quantities that are changing, both because it keeps explicit track of which derivative you took (“with respect to $x$”), and because it emphasizes that derivatives are ratios. Among other things, this helps you get the units right; mph are a ratio of miles to hours!
Both of these notations distinguish between dependent quantities ($f(x)$ or $y$) and the independent variable ($x$). However, in the real world one doesn't always know in advance which variables are independent. We therefore go one step further, and express derivatives in terms of differentials.