What is the point of reading classics over modern treatments? There seems to be a good number of mathematicians who recommend reading "classic" works in a given field (where the term "classic" is in the sense defined below). Indeed, there are many well written classic texts, for instance

*

*anything by Milnor, including some papers like "Group of Homotopy Spheres, I."

*Weyl, The concept of a Riemann surface

*Pontryagin, Topological Group

By "classic" text I loosely mean an old text written by the person who came up with the theory/idea or at least has made a major contribution somewhere very close to the subject of the book.
From a practical viewpoint, there are both advantages and disadvantages for reading the classics. Here are some that come to mind:
Pros:

*

*Reading the text gives a view of how the theory was originally created. (Thus by observing this, one may try to emulate the master's creative mindset.)


*By knowing that one is learning the theory "from the horse's mouth," one can have a peace of mind.
Cons:

*

*The terminology may be archaic. (e.g. A Course in Pure Mathematics by Hardy)

*The exposition may be presented in a cumbersome manner in view of modern machinery.

In particular, the "Cons" listed above result in more work and time spent on understanding the material than when reading a more modern textbook.
Edit: Per helpful comments (by @Geoff Robinson and @Francesco Polizzi) I am going to change my question to the following:
My Question. What are things one might learn from reading the classics that one would not gain from modern treatments?
 A: I don't know about mathematics, but in science it can very useful to read the classic papers which were produced about the time that the ideas which you are learning came into being.  This is often much better at giving someone a feel for the subject and how to actually do the subject, as opposed to being fed information via a very modern, efficient, streamlined teaching approach, which is not necessarily that good for giving people a feel for the way that the subject is actually done in practice.
Also I will just quote this from the Wikipedia article on Vladimir Arnold:
''He liked to study the classics, most notably the works of Huygens, Newton and Poincaré, and many times he reported to have found in their works ideas that had not been explored yet.''
A: As Alexandre said, one can't answer this question with sweeping generality, and a more case-by-case approach might be more appropriate. I think the most important item is missing from the list of pros. If $M$ is a modern rendition of a body of classical literature $C$, one would expect that a certain proportion $p$ of the content of $C$ would be accounted for in the treatment given in $M$, together with reformulation in modern language, putting things in the context of current research areas, etc. The OP seems to assume that $p=1$ always. Unfortunately, this is not so and it depends on the particular pair $(C,M)$ one is talking about. Hence the need for a case-by-case discussion.
If $p$ is substantially less than $1$, by only relying on $M$ and not making the effort to read $C$, one is not just missing a different perspective, but rather definitions, theorems and proofs. Areas where I think this is relevant are 19th century invariant theory, elimination theory, and even more so the intersection of the two subjects. A concrete problem at this intersection is to study representations of resultants in terms of the classical symbolic method, i.e., as combinatorially explicit contractions of tensors. For instance, in the BAMS 1984 review by Kung and Rota "The invariant theory of binary forms" they state this as an open problem:

However, explicit solutions in the case of binary forms were given by Gordan 1906 for the equal degree case and Dixon 1910 in the general case. So somewhere along the way, there was loss of content or $p<1$.
A: The question implies that there are pros and cons to reading classics, but only pros to reading the modern treatment. This seems contentious.
For instance 'The exposition may be presented in a cumbersome manner in view of modern machinery.' might seem to make sense, but it requires you to learn the modern machinery. Is the correct way to talk about orientability of a manifold to talk about homology with local coefficients? Maybe. If a beginner asks what orientability means, do you start by defining modules over group rings over fundamental groups? I think not.
Similarly, 'The terminology may be archaic.' is only a critique if you know all the modern terminology and none of the classical. Maybe Weyl's Riemann Surface book is useless to algebraic geometers of today, but I don't really know what a scheme is, so many modern books are useless to me (in fact, it often goes the other way, if there's some modern math I don't understand I try to find an old master who wrote before/during the time the terminology is being hashed out).
A: IMHO, everything depends on how the classical text (C) and modern text (M) are written. I will not talk about "archaic terminology", "outdated notation", etc. One skill that one has to acquire as early as his or her student years is to be able to translate any coherent exposition from one format to the other (visual to symbolic and back is usually the hardest one for most people to master; just different naming and outdated words should present no problem whatsoever), so just read C adapting it to the standards of the M language in your head (like most people read Shakespeare translated into relatively modern English and lose next to nothing). I will assume that you do not read just for the pleasure of reading, but have a single objective in mind to learn something new. Otherwise the question becomes "Do you like reading old books or not?" and the answer to that is "My personal preference is ..., period." I will also assume that by M, we mean a decent textbook covering the same subject as C, not a latest research article that starts with something like "it is well known (see [A],[B],...,[W]) that the gimbling of the slithy toves in the wabe is the primary cause of the Tumtum tree cross-contramulgation" to exclude the comparisons Ryan is talking about.
So really there are two (extreme; the real situation is often somewhere in between) possibilities:

*

*C is written in a hard to comprehend way (like in the Besicovich famous quote that "the pioneering works are always ugly, so the reputation of a mathematician is determined by the number of badly written articles [they have] published") or is not really rigorous enough and M runs on the same ideas but is just better structured, formalized, and streamlined in a few places. If so, read M and leave C to the historians to play with. All you can get from reading C in this case is a totally unjustified feeling of the superiority of modern ways and language over the predecessors' ones if you lack common sense and the appreciation of the efforts that went into the development of the modern style if you have one. The purely mathematical gain is zero either way.


*C is reasonably well written and has some ideas that were later superseded by alternative treatments presented in M (like, say, the modern treatment of the maximal function estimates is almost always via the covering theorems and the original Hardy-Littlewood one was rather via the rising Sun lemma and decreasing rearrangements). Then you may really want to read C along with M to learn the ideas that are no longer present in the mainstream and that is what you really want to extract from there and preserve in your memory.
Of course, it would be even better (for the purpose of the information absorption efficiency; the word "better" has no meaning without an objective modifier) if somebody else had done that extraction and presented it in the modern language. That is what I really expect from the books in "history of mathematics". The dates, priority questions, and the peculiarities of the personal lives of the main characters do not interest me, so, to my taste, one of the best history of mathematics articles is a short chapter in Littlewood's "Miscellany" about how the computation of the position of Neptune was done compared with how it could be done most efficiently. Alas, very few history books satisfy this criterion, so often one has to do this extraction work by him- or herself.
Just my 2 cents :-)
A: Some "worked examples" where re-reading the classics has proved worthwhile:

*

*Darboux's method for solving second order partial differential equations (per Goursat's 1896 book Leçons sur l'intégration des équations aux dérivées partielles du second ordre, à deux variables indépendantes), combined with its group-theoretic formulation of Vessiot (1939, 1942), has been extended by Anderson, Fels, and Vassiliou (as explained by Anderson in https://youtu.be/KKljvg3yK1U).


*Lie's ideas about using continuous groups as a tool to solve
differential equations were revived and extended in the latter half
of the 20th century - refer the CRC Handbook of Lie Group Analysis
of Differential Equations, edited by N. H. Ibragimov


*Cartan's (1908) method of equivalence has proved to be valuable in solving a number of practical problems (per Shadwick's recent lecture - https://youtu.be/f3ATo79f1QA).  See also Raouf Dridi's thesis Utilisation de la méthode d'équivalence de Cartan dans la construction d'un solveur d'équations différentielles, https://tel.archives-ouvertes.fr/tel-00264288.


*The work of Drach (1898) and Vessiot (1946) has been revived, corrected and incorporated into the nonlinear differential Galois theory of Umemura and Malgrange. (See also Pommaret's 1983 book Differential Galois Theory.)
A: One cannot make sweeping generalizations here. I mean there is no general rule on what is better, to read classics or modern expositions. Both can be profitable for a modern research mathematician.
I recall that our business consists not only in proving theorems, but also in finding the right questions to work on. And the second part can be much harder. To understand which questions are important and which are not, one has to have an historic perspective of developing
of the subject. This thing is frequently lost in modern exposition many of which aim at the shortest and most economic way to prove things.
A: I cannot resist quoting the epigraph of the beautiful paper by Andreotti and Mayer, On period relations for abelian integrals on algebraic curves,
Ann.  Sc. Norm. Sup. di Pisa 21, no. 2  (1967),  189-238. It is an excerpt of the pamphlet "Il teatro alla moda" by the Venitian composer Benedetto Marcello (my translation):

In the first place the modern poet must not have read, nor read, the antique authors Latin or Greek. Indeed the antique Greeks or Latins have never read the moderns.

A: A question very similar to yours was asked quite a few years ago on MO here. Take a look at the answers there.
