In differential calculus we start with a function f(x) and we then find the relationship between f(x) and its various derivatives. Differential equations is the inverse of this process. That is we start with the relationship between f(x) and its various derivatives, from which we are required to determine what f(x) is.
For example, in differential calculus we start with f(x) = sin x + cos x, and then show that f''(x) + f(x) ≡ 0. In the study of differential equations we start with f''(x) + f(x) ≡ 0 and our task would be to determine what the function f(x) has to be.