TITLE: What is the point of reading classics over modern treatments?
QUESTION [25 upvotes]: 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?

REPLY [2 votes]: 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.)