This answer consists of me waving my hands, if you want to get a better understanding I would advise you to have a glimpse at any textbook in calculus of variation, like the book by Struwe. Or you might want to have a look at chapter 10 in Lieb & Loss "Analysis", where they explain how the calculus of variation is useful in quantum mechanics. This variational approach is used in classical physics as well and is called Hamilton's principle.
As mentioned in my comment above, calculus is usually done for functions from $\mathbb{R}^n$ to $\mathbb{R}^m$. On the other hand, calculus of variation typically aims to understand functions from some (infinite-dimensional) function space to $\mathbb{R}^k$.
While the definition of integration, differentiation and so on are quite similar to the one in calculus, there are always some technical difficulties that pop up in infinite dimensions. The most basic one is that linear functions no longer need to be continuous in infinite dimensions! See here for an example Discontinuous linear functional.
One of the things we care about in calculus is to find extrema of functions. Typically we proceed in the following way:
1.) Show that some extremum must exist. This is usually done using some sort of compactness. For example the function is continuous and goes to infinity for $\vert x \vert \rightarrow \infty$.
2.) We compute the critical points.
3.) We know that global extrema are critical points.
4.) We check all the critical points and the smallest one is the global minimum.
In infinite dimensions the first point becomes quite difficult as compact sets are kind of "small". For example closed balls are no longer compact Is it true that the unit ball is compact in a normed linear space iff the space is finite-dimensional?. One can the pass to different topologies where one does have compactness (the Banach–Alaoglu theorem is often quite useful), however, then one needs to ensure that the function in question is also continuous with respect to that new topology (typically this is not really true, but we can get away by showing that it is lower semicontinuous in that new topology).
Let me briefly comment on the critical points. In calculus, typically the set of critical points is relatively "small". However, in calculus of variation we usually have that critical points are characterized as the solution set of a PDE.
So while calculus of variation has similar goals as calculus and uses similar ideas it is substantially more technical.
