What is the difference between smooth manifolds and differential geometry? I am planning to study geometry (I know this is very vague, but I can't exactly explain what I mean, because I don't fully understand the distinction between various terms, which is the reason for this question).
I'm confused by titles of books. As I understand, there are books which have "Smooth Manifolds" in their title, such as Lee's book Introduction to Smooth Manifolds. On the other hand, there are books which contain "Differential Geometry" in their title, such as Spivak's book A Comprehensive Introduction to Differential Geometry.
Looking at the table of contents, I don't really understand the difference between  books on "smooth manifolds" and books on "differential geometry". Could someone explain what the distinction is, and perhaps suggest what is the usual order of studying them? In other words, would you first study something which has "differential geometry" in the title, or "smooth manifolds"?
Similar question about difference between Riemannian geometry and differential geometry
 A: In geometry, you can measure distances, angles, and areas. In topology (for example in smooth manifolds) you can't measure any of those.
Also, in differential geometry you generally have surfaces sitting in space, so you have extrinsic information, whereas with Riemannian geometry all you have is intrinsic information. (For example, a section of a cylinder is intrinsically the same as a section of a plane because you can flatten it out without stretching anything, even though extrinsically one's curved and the other isn't).
I would recommend differential geometry first because it's more concrete.
Personally I did Riemannian geometry first out of the three, through do Carmo's book (self-study), but it has a "chapter 0" on a smooth manifolds / differential topology refresher that took me some time to bang my head against
A: Smooth manifolds are like the background things you work with: the base layer/ingredients of a cake (eggs, flour etc). Geometry is the study of things like lengths, angles etc on smooth manifolds (like extra ingredients on a cake: chocolate, vanilla etc, each type of geometry having a completely different flavor, and techniques).
The stuff in Lee's Introduction to Smooth Manifolds book is indeed very similar in topics to say Spivak's volume 1. Now, regarding the name, here's a part quoted directly from Lee's preface to Introduction to Smooth Manifolds:

This subject is often called “differential geometry.” I have deliberately avoided using that term to describe what this book is about, however, because the term (referring to "differential geometry") applies more properly to the study of smooth manifolds endowed with some extra structure—such as Lie groups, Riemannian manifolds, symplectic manifolds, vector bundles, foliations and of their properties that are invariant under structure-preserving maps. Although I do give all of these geometric structures their due (after all, smooth manifold theory is pretty sterile without some geometric applications), I felt that it was more honest not to suggest that the book is primarily about one or all of these geometries. Instead, it is about developing the general tools for working with smooth manifolds, so that the reader can go on to work in whatever field of differential geometry or its cousins he or she feels drawn to.
(emphasis mine)

If you just want to get a feeling for what the subjects are about, I think reading the prefaces of these various books (Lee has 3 books, Spivak has 5 volumes (the first two being most introductory/"important")) is a good start.
As an example other than Lee and Spivak, here is the preface (in full, because this text is harder to find) to Dieudonne's Treatise on Analysis, Vol IV, Chapter XX (20), titled "Principal Connections and Riemannian Geometry", describing roughly what Riemannian geometry is about, and some of its history:

The “naive” concept of differential geometry regards it as the study of curves and surfaces in physical space $\Bbb{R}^3$,and more generally of “manifolds ”embedded in real space $\Bbb{R}^N$.For a long time the development of the subject centered on the metrical notions which were regarded as “natural” in $\Bbb{R}^N$, and which on “curved” submanifolds gave rise to the classical notions of length, area, volume, etc. This point of view emphasized the role of the group of Euclidean displacements, under which these notions are invariant, and two submanifolds of $\Bbb{R}^N$ which are related by a displacement have always been regarded as intrinsically the same. From the inception of infinitesimal calculus, it was realized that one could attach to each point of a plane curve (for example) a number depending on the point which measured the “curvature” of the curve at that point, in the intuitive sense of the word; and it is easily shown that the knowledge of this number as a function of arc length determines the curve uniquely, up to a displacement. This is the starting point of the study, pursued unremittingly over two centuries, of the “differential invariants” of manifolds embedded in $\Bbb{R}^N$:a study which for a long time remained purely “local,” but which from the beginning of this century came to include many “global” problems concurrently with the development of topology.
Superimposed on this line of development, since the time of Gauss, has been the “ intrinsic” conception of manifolds equipped with an “infinitesimal” metric, independent of any embedding in real space $\Bbb{R}^N$. It required the efforts of Riemann and several subsequent generations of mathematicians to lay the foundations of this new theory, with the help of what was called “tensor calculus” (which is nothing more than “localized” multilinear algebra). Other efforts were necessary to extract and separate out the concept of a differentialmanifold (not endowed with a metric) from that of a Riemann- ian manifold: on a Riemannian manifold, as was shown by Levi-Civita, the metric defines a unique “connection” (in the sense of (17.18)), which generalizes the intuitive idea of "parallelism” in $\Bbb{R}^N$ (20.9).
It appears at first sight, when one embarks on the study of general Riemannian geometry, that group theory ceases to play a part (in general, a Riemannian manifold possesses no metric-preserving diffeomorphism other than the identity). A fundamental advance, due to E. Cartan, was the recognition that Lie groups play as important a role in differential geometry as in classical geometry (in the sense of Klein). By refining and developing the ‘‘method of moving frames,” which had proved its usefulness in the classical theory of surfaces, he showed that vis-a-vis an arbitrary Riemannian manifold $M$ (or rather its tangent bundle $T(M)$), there is a principal fiber bundle, whose group is the orthogonal group, which plays a role entirely comparable to that played by a Lie group vis-a-vis its homogeneous spaces. Here again, the technique of "lifting everything up to the principal bundle of frames” reveals best the nature of problems in differential geometry (cf. Section 20.5).
This method need not be restricted to the orthogonal group, and accord- ing to the nature of the group $G$ of the principal bundle under consideration, we obtain different geometries ("$G$-structures") in great variety. It has been impossible within the framework of this Treatise to do more than indicate the general principle which leads to these “geometries” (Sections 20.1 to 20.7), and we must refer the reader to specialized monographs for informa- tion on spaces endowed with an affine connection, a projective connection or a conformal connection, Finder spaces, Hamiltonian structures (or contact structures), and their relations with dynamics, etc. (see [37, 54-56, 59, 61, 63, 70, 711]). The second part of the chapter is devoted to Riemannian manifolds; but here again the subject is so vast that we have regretfully had to pass by in silence many parts of it. The great theory of Riemannian symmetric spaces ([45, 62, 661) is hardly touched on; the fundamental notion of the holonomy group is not mentioned, nor is the geometry of Hermitian manifoldsandtheir most important particular case, the theory of Kahler manifolds ([18,45]). Our study of geodesics does not go beyond the level of the most elementary theorems, and although this study constitutes one of the most attractive and complete chapters of the calculus of variations, this latter theory also remains beyond our horizon: both as regards the study in depth of extremal conditions in the case of simple integrals and the “multidimensional" general theory ([53, 58, 63, 65, 68]), and as regards the most beautiful application to analysis and topology of I-dimensional calculus of variations, namely Morse theory ([63, 671).
Although the techniques of analysis on manifolds which we have developed in Chapters XVI to XVIII underlie the results of this chapter, nevertheless it has a much more pronounced “geometrical” flavor than the other parts of this treatise, and corresponds to what used to be called “the application of analysis to geometry.” However, in accordance with the spirit of modern mathematics, Geometry pays back in large measure the support it has drawn from Analysis, as our allusions above indicate, and as the reader will see for himself repeatedly in later chapters.

Obviously you don't have to udnerstand everything here; just gain whatever nuggets of inspiration/information you can and move on (for now:)
