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The wikipedia articles on differential calculus and differential geometry are quite long and not so straightforward for a layman like me. Is there a master of math vulgarization out there that could summarize in couple strong sentences what differential calculus is and how it is linked to geometry?
I have read this nice post and this one too which give good examples and references but does not really explain what "differential" means, this post is good too but focuses on equations.
Here is my understanding : first, differential calculus is a toolbox that allows one to actually compute things in differential geometry.
Differential geometry is a way to describe geometric objects with more precision that for instance topology : in differential geometry, we can make sense of such notions as distance and curvature, which we cannot (or in the case of distance, not so precisely) in usual topology. More importantly, since every bit of our space is linked to a piece of some euclidean space, we can use the tools of differential calculus to describe precisely what "speed" or "acceleration" means; first locally, through maps, then globally, by gluing maps together appropriately.
Now very often, the "geometric objects" we study are in fact phase spaces (spaces of all possible positions and speed of a certain number of physical objects), that are seen with a geometric eye, because it allows to use the tools described earlier to predict the evolution of a physical system.
Concerning the word "differential", see it as a pointer to mean that we study the evolution of the really small differences through functions. For instance, the derivative of a single-variable funtion $f:\mathbf{R}\rightarrow\mathbf{R}$ tells us that a very small difference in the variable $x$ in the source will become $f'(x)$ times this difference in the range, if our difference is "infinitesimally small". In other words, to relate to what I wrote earlier : the "small difference" may be seen as the "speed of variation". The "differential" stuff tels us not only how the points ar afected by functions, but also how the "speeds of variation" are modified.
I hope this helped. I don't know what background you have, but if you are say a 2nd year student in university, I would recommend "Introduction to Smooth Manifolds" by John M. Lee. I have a very geometric eye, and this book appealed to me.
Euclidean geometry provides equations between lengths, areas and (trigonometric functions of) angles in figures containing straight lines and circles, including tangents to circles. Differential geometry in the plane provides equations containing in addition to the above lengths of arbitrary smooth curves, angles between tangents of such curves, and finally allows to talk about the curvature of smooth curves.
A typical theorem of plane differential geometry is the following: The total turning angle of the tangent vector to any simple closed curve in the plane (meaning: smooth and no self intersections) is $\pm2\pi$ (and not $2k\pi$ for some $k\ne\pm1$).
