History of the construction of $\mathbb{R}$ When did the constructions of Reals take place?
What is the latest construction the one due to Cantor (by Cauchy Sequences) or the Dedekind?
I ask because the  trustful reference (baby Rudin) that I looked,  tells they were published in the same year 1872. 
That seems to me, Cantor's idea was more fruitful (however was quite similar) because it was used to complete any Metric space. Nevertheless the Dedekind construction may rest as an Historical thing when the Cantor idea should be the one taught obligatorily in the Analysis courses.
Personal commentary: I don't  think the Dedekind's idea can be used in any other construction.
 A: The only reference I can give is Akihiro Kanamori's historical overview in The Higher Infinite, in which he cites papers by Dedekind and Cantor for their constructions - both given in 1872.
To remark on your personal comment, I have to say two things (which are way too long to fit into a comment):

*

*The fact that Cantor's construction can be carried out in a general metric space has one caveat, as one could argue that it is simply the same thing in the case of $\mathbb R$. However it is not: in the general case we use the completeness of $\mathbb R$, while when constructing $\mathbb R$ we do not yet know that it is complete. The proof of its completeness is not very hard, but it is something one needs to do nonetheless.


*Dedekind's construction is actually very useful. If we take a partial order $(P,\le)$ we can embed it densely in the set of its cuts, $L_p=\{x\mid x<p\}$ ordered by inclusion. This gives us a complete partial order, that is every bounded set has a least upper bound.
If we started a separative partial order then the result is a complete Boolean-algebra in which $P$ is embedded densely. The original separative partial order could have been a Boolean-algebra to begin with, which will then yield the completion of a Boolean-algebra.
This quite useful in forcing via Boolean-valued models, and it can also help us transfer a problem from partially ordered sets into Boolean-algebras.
Indeed the Dedekind construction has its own uses. You may have said what you said simply because one do not run into Dedekind completions very often outside rather set-theoretical settings (not just set theory, but fields close to it) which is not often appearing in undergrad or even grad level studies.
It is also worth noting that from Dedekind's construction it is a lot easier to prove that every bounded subset of $\mathbb R$ has a supremum, and it is very easy to find this supremum as well.
A: As to what is the most recent construction of the reals, I can't say for sure. I think it is not too fashionable to come with new constructions, but one can construct more general number systems that contain the reals as a subset‚ without constructing the reals first. One nice and fairly recent example is Conway's surreal numbers, first constructed around 1970, I think. Donald Knuth's book Surreal numbers is a quite readable exposition, a bit like a mystery novel.
A: Here are some highlights, concerning mostly the irrationals, from chapter 41 of Morris Kline's Mathematical Thought from Ancient to Modern Times.
Euclid had a notion of "incommensurable ratios", which Kline argues are just the irrationals from a different point of view. Euclid also had the notion of defining equality of incommensurable ratios by, given one of these ratios, dividing the rational numbers into two classes, those for which the rational is less than the incommensurable ratio, and those which are greater. This reminds one of Dedekind cuts; a fact which Dedekind himself acknowledged.
William R. Hamilton offered the first (incomplete) treatment of irrational numbers  in two papers read before the Royal Irish Academy in 1833 and 1835. He also had a notion of Dedekind cuts.
Cantor pointed out that the previous work tried to define the irrationals as limits of rationals, whilst the limit, if irrational, is not  defined logically unless the irrationals are already defined. At this time, 1859, Weierstrass gave a theory of the irrationals. This was supposedly published in Die Elemente der Arithmetik in 1872 by H. Kossack; though Weierstrass disowned the work
In 1869 Charles Méray gave a definition of the irrationals based on the rationals.
George Cantor gave his theory in 1871.
Eduard Heine gave his theory in 1872 in the Journal für Mathematik (74, 172-178).
In the same year Dedekind gave his theory in Stetigkeit und irrationale Zahlen (3, 314-334).)
After all this, the rational numbers were put on a rigorous basis, starting with the integers, with works of Dedekind in his Was sind und was sollendie Zahlen(1888, 16, 335-391) and, more notably by Peano with his axiomatic approach in  1889 in his Arithmetices Principia Nova Methodo Exposita.
A: To complement the fine answers given earlier, I would just point out that the first construction of the reals was arguably given considerably earlier than the above discussion suggests.  Namely, Simon Stevin already envisioned representing each number by an unending decimal, and emphasized that there is no difference between rational numbers and other numbers (such as surds) in this respect. Arguably, the reason mathematicians like Cauchy did not bother "constructing" the real numbers is because they felt such a number system had already been constructed, and used for centuries. More details can be found in this article on Stevin.
