This difference was well-known in the 19th century when people
a) Knew about invariants, and
b) Calculated by hand.
I believe a lot of the confusion today stems from Lang's Algebra book which is at best misleading about how to interpret what the resultant and discriminant are (and the ideas of famous books, right or wrong, tend to be perpetuated in other people's books!).
As an example, the resultant of the two polynomials $3x+1$ and $3x+2$ is, according to Sylvester's matrix definition, equal to $3$. Here Voloch's $D=1$. Surely this makes no sense according to the well-known theory, that a prime $p$ dividing the resultant of two polynomials should be interpreted to mean that these polynomials share a root when reduced mod $p$? This is evidently nonsense in this example ... unless one re-interprets these polynomials projectively (which is what one should do).
But now if we look at the pair of polynomials $y+2$ and $3y+2$ then the resultant is $4$, and here $D=4$, but how do you interpret here $2^2$ dividing the resultant? It is not immediate from the interpretation of modern algebra books!
There are all sorts of reasons that prime powers can divide a resultant (and discriminant) and it is complicated to understand all the cases when you wish to interpret higher power divisibility.
In Bhargava's work, he needs to understand squarefree values of a multivariable polynomial which is the the value of a discriminant of a class of parametrized polynomials. In other words he needs to parametrize when $p^2$ divides terms in this particular class of discriminants. Even this relatively simple request breaks down into several non-trivial cases, which he handles so beautifully as if to make it look trivial, but it's not.