In implicit differentiation you are taking a known expression, say of two variables x and y, that cannot be expressed as a function y = f(x), and you are differentiating it with respect to one of the variables, say x, in order to solve for the derivative of the other variable, dy/dx.
For a simple example, an equation of a circle is ##x^2 + y^2 = 1##. This cannot be put in the form y = f(x) for a single f(x) that describes the entire curve. If we differentiate this with respect to x, we find
$$2x + 2y \frac{dy}{dx} = 0 \Rightarrow \frac{dy}{dx} = -\frac{x}{y}.$$
Note that we started with a specific expression that related x and y and derived dy/dx.
In a differential equation, you are given an expression in terms of the dy/dx, y and x and asked to find the function y = f(x) (or even an implicit relation between the variables) which satisfies this equation.
So, for example, say you were just given the equation dy/dx = -x/y. You don't yet know anything beyond this equation. When you solve this equation, you find that ##y^2 = -x^2 + C##, for some constant C. (You may be given some initial data which enables you to pin down this constant, e.g., y = 1 when x = 0 would specify C = 1).
Again, the difference here was that we had an equation for dy/dx given in terms of x and y, and we had to solve for the relationship between y and x that satisfies that differential equation.
In the first case, we had the relation between x and y, and we wanted to compute the derivative dy/dx.