If we have two circles in the plane described by $(x_1, y_1, r_1)$ and $(x_2, y_2, r_2)$ we can determine if they are completely disjoint by simply:
$$(x_1 - x_2)^2 + (y_1 - y_2)^2 < (r_1 + r_2)^2$$
Assume this is not the case, we now want to know if one completely overlaps the other. (That is: if the second circles interior is a subset of the first circles interior.)
If $(x_1, y_1) = (x_2, y_2)$ than we can trivially compare radii, so lets assume their centers are distinct.
The way I have imagined is to create a parametric equation of the line that connects the two centers:
\begin{align} x_p(t) &= x_1 + t(x_2 - x_1) \\ y_p(t) &= y_1 + t(y_2 - y_1) \end{align}
Then we calculate the two line segments (represented as two pairs of t values) where the circles intersect this line. One is a subset of the other if and only if the corresponding circles are subsets of each other.
Is there a simpler approach I am overlooking?
The furtherest points between a circle and a point, would be on the line from the point, through the center of the circle.
Hence, circle 1 is contained in circle 2, if and only if circle 2 contains that furtherest point, which implies that
$$ r_2 \geq \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2} + r_1 $$
As such, one circle will be contained within the other if and only if
$$ (r_1-r_2)^2 \geq (x_1-x_2)^2 + (y_1-y_2)^2$$
Note: This approach should be similar to how you show that 2 circles are disjoint (since the 2 nearest points again lie on the line connecting the circles).