What is the application difference between extent and solidity in image processing?
I would suggest that Solidity is the better measure. Extent would be a cheap approximation. It’s cheap because computing the bounding box is cheaper than the convex hull (and really simple to implement).

Solidity is sometimes referred to as Convexity, but more commonly Convexity is the ratio of the perimeter of the convex hull to the perimeter of the object. These two measures are of course related, but not the same.

Solidity doesn’t quantify roundness. A rectangle has a Solidity of 1 (the maximum posible value). And so does a triangle. Roundness (or similarity to a circle) can be conveniently computed by comparing the area and the square of the perimeter, or the area to the product of the width and height (look for Feret diameters to compute these). Another approach is to determine the coefficient of variation of the distance of each boundary pixel to the geometric center of the object.

Solidity is useful to quantify the amount and size of concavities in an object boundary. Holes are also often included. For example, it distinguishes a star from a circle, but doesn’t distinguish a triangle from a circle. 

Here is a good survey of object shape features: M. Yang, K. Kpalma and J. Ronsin, "A Survey of Shape Feature Extraction Techniques", in: Pattern Recognition Techniques, Technology and Applications, P.Y. Yin (Editor), I-Tech, 2008.

The DIPlib library implements the more meaningful of these features. (Disclosure: I implemented most of those features in DIPlib.)