Why is differentiation called differentiation?

The derivative (differential) is defined as the limit of the difference quotient

$$f'(b) = \lim_{a \to b} \frac{f(b) - f(a)}{b-a}$$

where difference quotient refers to the difference of $f(b)$ and $f(a)$ in the numerator and the difference $b$ and $a$ in the denominator.

The derivative is also defined (per Leibniz) as the ratio of differentials $dy$ and $dx$,

$$\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

where $dy$ and $dx$ represent infinitesimal changes (differences) in $y$ and $x$, respectively.

As far as history of the term goes, differential was coined by Gottfried Leibniz as described here.

1684 G. Leibniz Acta Eruditorum 3 469 Ex cognito hoc velut Algorithmo, ut ita dicam, calculi hujus, quem voco differentialem, omnes aliae aequationes differentiales inveniti poſſunt per calculem communem, maximaeque & minimae, itemque tangentes haberi

[Just by knowing the algorithm, as I call it, of this method, which I call differential, all other differential equations can be solved by a common method, and maxima and minima, and tangents too, can be found]

Isaac Newton used the notation $\dot{y}$ to denote the generated rate of change in $y$, which he called a fluxion. Leibniz's notations are generally what are used in calculus today, though Newton's dot notation is still sometimes used for derivatives with respect to time, particularly in physics.

The etymological root of "differentiation" is "difference", based on the idea that $dx$ and $dy$ are infinitesimal differences.

If I recall correctly, this usage goes back to Leibniz; Newton used the term "fluxion" instead.