Broadly speaking, a book's age does not serve to credit or discredit it. That being said, it's a bit like asking, "Is a Magnavox Odyssey still valid"? If it still plays, then it still plays, but you also have to deal with the fact that old things are meant to do different things than new things, and that even if two things have the same goal, time will still help to refine that thing through innovation.
Will Hardy's analysis be correct? Most likely. But you run into two things, especially when you're talking early twentieth century books. For one, the language will be very unorthodox by today's standards. If you're going particularly early in this period then you might even find dissent among authors as to what to actually call what we would refer to now as a "set" (I believe Russell had used "manifold" at some point for what is now a set, and now manifold has a very particular meaning in geometry). Be prepared to Google what words mean, and be prepared for that Googling to be a nontrivial endeavor. It's also important to note that an important part of mathematics is knowing how to read and communicate it, neither of which will be possible if your lexicon is a century old.
Secondly, their methods and approaches will probably not be what we'd use today. This is the refinement. Over time, mathematicians will look for better ways to do what the older folks did. It's common that a classic theorem's proof when first presented will consist of multiple lemmas, and will be a long and drawn out labor, while later authors will "streamline" and "refine" those methods. So though what folks like Hardy might give you is in all likelihood correct, it's also likely that later authors will have improved upon what the classics present. And when I say "improvement", I don't mean that some stuffy journal has found some generalization of a theorem to the point that it's almost unrecognizable; I mean that your typical undergrad text will do in six lines what an older author did in a page and a half, and probably in a reasonable generalization.
EDIT: This newfound brevity might also sometimes be a mater of saying something very similar to what the older authors were saying, but simply having more precise language to say it with.
Best Answer
The second book you suggest, A Course in Calculus and Real Analysis, is maximal in the set of Calculus books partially ordered by quality.
In other words there is no better book, although there might be excellent books which cannot be said to be better or worse than this one.
Its main qualities are:
The incredible quality of the colour illustrations which almost evoke an art book.
For example, look at the weird piecewise linear zigzag function on page 27.
The Notes and Comments at the end of each chapter, which really explain what is going on under the hood and give extremely interesting information on the history of the topic discussed in the chapter.
Each of these topics is discussed in depth and many results are quite interesting and unusual in a textbook, as for example on page 78 the discontinuous function having the intermediate value property.
Another result (not trivial at all despite appearances) that I have never seen discussed in a Calculus textbook is that the logarithm is a transcendental function: the proof is found on pages 270-271.
And as an algebraic geometer I was astounded and delighted to find on pages 143-144 a discussion of the multiplicity of a plane algebraic curve at one of its points: I am not aware of any Calculus book having ever discussed that subject!
The many exercises (around 500), some quite challenging and/or instructive.
For example in the exercises to Chapter 9 you can learn about Raabe's test and the hypergeometric series.
And what about the other book, A course of Pure Mathematics ?
It was written in 1908 by Hardy, the best 20th century British analyst, and helped revive Analysis in Great Britain, a country that lagged far behind the continental school in that subject.
Britain was brilliant in Algebra, Geometry and above all Applied Mathematics, but was no match for the illustrious analysts across the Channel: Weierstrass, Riemann, Volterra, Hermite, Cauchy, Jordan, de la Vallée-Poussin and dozens of others.
Hardy wanted to open his compatriots to modern Analysis and a stepping stone was the book under discussion, written in the style of "a missionary preaching to cannibals", as he himself wittily acknowledges in the preface to the seventh edition.
I love that book for its beauty and depth, but also for nostalgia's sake.
To sum up Hardy's is a great book to browse and enjoy, but for a young mathematician wanting to learn the subject pleasantly but in depth I warmly recommend A Course in Calculus and Real Analysis.