calculus - How is the modern definition of derivatives different from that of Leibniz' "infinitesimal"-based system?

I pose this question knowing fully well a somewhat similar question was asked in the popular question Is $\frac{dy}{dx}$ not a ratio?

However, as great as the answers are, the answers stop short of explaining a particular culprit I run into every time I try to understand the difference between the modern definition, and that of Leibniz. As such, at least for my own sake and possibly that of others, I feel the need to ask this question.

Derivative picture

The problem I'm having is that when I look at the page on derivatives in my calculus book, I see the all-familiar drawing detailing how to think about the limit definition of derivatives, as pictured above. This is supposedly different from Leibniz' idea of the ratio of two infinitesimal quantities - but I don't understand how.

In both cases, we have a Δy being divided by a Δx. In Leibniz' vision, Δy becomes dy and Δx becomes dx, two "infinitesimally small quantities", smaller than anything imaginable but still greater than zero. In the modern limit definition, Δy becomes dy, an unimaginably small quantity, and Δx becomes dx, again an unimaginably small quantity. That, to my untrained perspective, looks identical to Leibniz' idea of derivatives, except the concept of dy and dx being "infinitely small" is now embodied in the limit.

How is this modern definition any different from Leibniz' definition that relies on the "ghosts of departed quantities"?

Edit: I should emphasize that I'm not looking for answers through the lens of non-standard analysis. This is a regular, standard calculus question.