When one is first introduced to a differential of a real-valued function, it is defined as a linear function that takes tangent vectors to real numbers. That is, differential is a linear functional on the tangent space at a point, where tangent space is thought of as the space of all arrows emanating from the point (see Mathematical Analysis I by Zorich, for example).
It is straightforward to show that $df = (D_if)\,dx^i$, or, more formally, $df(h) = (D_if)\,dx^i(h)$. However, when dealing with smooth functions, it is possible to identify this geometric tangent space with the space of derivations at a point and call it the tangent space. Then we define the differential, which is now called a one-form, as a linear functional on this newly identified tangent space by $df(X) = Xf$ (see Introduction to Manifolds by Tu, for example). It is true then that $df = (D_if)\,dx^i$, or, yet again being more formal, $df(X) = (D_if)\,dx^i(X)$.
Then many authors say that one-forms are just the differentials. But I don't quite get it. Of course, differentials and one-forms act on isomorphic spaces, thus they are also isomorphic. But still, they act on different spaces. Because there is an isomorphism between them, structurally one-forms and differentials are completely analogous. But how can they be PRECISELY the SAME thing when they act on different spaces? Wouldn't it be more correct to say that every one-form can be identified with a unique differential?
As Moishe Kohan has stated in the above comment, Not every 1-form is the differential of some function. A 1-form $w$ would be defined as smooth section of the cotangent bundle. When described locally at a point on your manifold, $w$ can be decomposed with respect to the basis of the cotangent space thusly;
$$w_{p}=f_i(p)dx^i$$
Where $f_i\in C^{\infty}(M)$.
Now, the differential of some smooth function, $g\in C^{\infty}(M)$, would be;
$$dg=\frac{\partial g}{\partial x^i}\bigg|_{p}dx^i$$
So if every 1-form,$\space w$, was the differential of some smooth function then we would be able to express every $f_i$ as the derivative of the same function $g$ with respect to a given chart component $x^i$. This is clearly not always possible. In general, the space of 1-forms is much larger than the space of differentials of smooth functions.