I confess, in 1965 or so, I first thought that the version of the SvKT (Seifert-van Kampen Theorem) for the fundamental groupoid enabled us to get rid of base points. But then I wanted to calculate the fundamental group of the circle, and gradually realised that we needed $\pi_1(X,A)$, the fundamental groupoid on a set of $A$ of base points chosen according to the geometry. Here is an example of the kind of situation which the standard texts do not give:
and for which covering space methods are not ideal. In fact one needs combinatorics and combinatorial group(oid) theory to calculate individual $\pi_1(X,a)$ from $\pi_1(X,A)$. See my book Topology and Groupoids and also the classic 1971 book (downloadable) by Philip Higgins, Categories and Groupoids.
Groupoids model homotopy 1-types. So one first determines the 1-type before calculating an individual fundamental group.
This idea is nicely modelled in higher dimensions: one can calculate some 2-types and higher types as "big" algebraic objects inside which are the homotopy groups which one might want. The methods are explained more in a talk given in Paris on June 5, 2014, at the IHP, available on my preprint page.
As Mariano points out, this has been discussed elsewhere on stackexchange and mathoverflow.
July 14: The most general version of the SvKT is given in  (downloadable) on my publication list,
R. Brown and A. Razak, `A van Kampen theorem for unions of
non-connected spaces'', Archiv. Math. 42 (1984) 85-88.
in the form of a coequaliser statement when given an open cover and a set $A$ of base points which meets each path component of each 1-,2-, and 3-fold intersections of the sets of the cover. The style of proof goes back to Crowell's original version, and has the advantage of generalising to higher dimensions. For example, in
 R. Brown and P.J. Higgins, ``Colimit theorems for relative homotopy
groups'', J. Pure Appl. Algebra 22 (1981) 11-41.
The retraction from the version for the full fundamental groupoid, $A=X$, to this version is quite difficult to manage and is done in May's "Concise..." book, without the refinement on the conditions, but I feel is the wrong way to go, though it is quite elegant for the pushout version in dimension 1.
July 17, 2014: Actually I have missed out three reasons why one is interested in the fundamental groupoid $\pi_1 X$.
The notion of fibration of groupoids is relevant to topology particularly in construction of operations of groupoids on homotopy sets, and exact sequences. If $p: E \to B$ is a fibration of spaces then $\pi_1 p: \pi_1 E \to \pi_1 B$ is a fibration of groupoids. This is exploited in Chapter 7 of Topology and Groupoids. See for example arXiv:1207.6404 for other current uses of the algebra of groupoids, and in particular fibrations.
Similarly if $p: E \to B$ is a covering map of spaces then on applying $\pi_1$ we get a covering morphism of groupoids. Thus a map is modelled by a morphism, and this often makes the theory easier to follow (IMHO!), particularly with regard to questions of lifting maps. See Chapter 10 of T&G.
If $G$ is a (discrete) group acting on a space $X$ then it also acts on the fundamental groupoid $\pi_1 X$. So we have not only orbit spaces $X/G$ but also orbit groupoids $(\pi_1 X)/\!/G$. There is a canonical morphism $(\pi_1 X)/\!/G \to \pi_1(X/G)$ and there are useful conditions which ensure this is an isomorphism, e.g. $X$ is Hausdorff, has a universal cover, and the action is properly discontinuous. See Chapter 11 of T&G. One example given there is the cyclic group $Z_2$ acting on $X \times X$ whose orbit space is the symmetric square of $X$. Under useful conditions, its fundamental group is that of $X$ made abelian. It would be good to see lots more examples.
Aug 17, 2016: I can now refer to my answer to my own mathoverflow question on the relation of the van Kampen Theorem to the notion of descent.
Oct 17, 2016: I should add that a whole area of research into the use of strict higher groupoids in algebraic topology, one part of which is described in the book Nonabelian Algebraic Topology, (EMS 2011, 703 pages), arose out of seeking generalisations to higher dimensions of the use of the fundamental groupoid.
December 23, 2016 It may be useful to point out that a new volume by Bourbaki "Topologie alg\'ebrique" Ch 1-4, (Springer) 2016, uses the fundamental groupoid extensively, and relates its use to descent theory. It does have results on orbit spaces, but no example of applications. It does not use the fundamental groupoid on a set of base points.
December 9, 2019 It should be remembered that homotopy groups of a pointed space were introduced at the 1932 ICM at Zurich, by E. Cech, but the idea was not welcomed because of their abelian nature, so that they did not seem satisfactory higher dimensional versions of the fundamental group. The idea was taken up by Hurewicz, and the fascination and utility of homotopy groups led to the idea of a nonabelian version being regarded as a mirage, though the early homotopy theorists were fascinated by the action of the fundamental group on the higher homotopy groups (a comment of J,H.C. Whitehead, 1958).
We now know that you can construct higher analogues of the fundamental groupoid for certain structured spaces, e.g. filtered spaces and $n$-cubes of pointed spaces. For the filtered case, see this 2011 book.
Note that the fascination and difficulty of the study of higher homotopy groups may, since these groups are defined only for pointed spaces, have deterred people from considering the possible uses of the many pointed case. Also the use of groupoids in algebraic topology has been in the past dismissed by many.
Nonetheless, there is the basic fact of life (to use a term of JHCW) that while "group objects in the category of groups are abelian groups", that is not so if the word "group" is replaced by "groupoid". In view of the importance of group theory in maths and science, it is reasonable to ask of the potential significance of that basic fact of life.