Important results with one or more than one proof
I believe the Atiyah-Singer Index Theorem qualifies as deep. There are several different approaches to proving it, see here.
The first example that occurs to me is Hindman's theorem: If the set of positive integers is partitioned into finitely many pieces, then there is an infinite set $H$ such that all sums of finitely many (distinct) elements of $H$ lie in the same piece. Hindman's original proof is very complicated (Hindman himself has suggested that it could be used to torture graduate students) but it has the advantage of being elementary --- it can be formalized in a system only slightly stronger than $ACA_0$. A later, easier proof by Galvin and Glazer, now considered the standard proof, has the advantage that one can easily remember or reconstruct it, but it requires more powerful tools, including an application of Zorn's lemma to a collection of subsemigroups of a certain semigroup whose elements are ultrafilters. There's also an "intermediate" proof due to Baumgartner.
The Mordell conjecture (proved by Faltings (2x), Vojta and Bombieri).
The Weil conjectures (proved by Deligne (2x)).
The Theorem of Roth (proved by Roth and Faltings).
References:
Faltings, Gerd (1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" [Finiteness theorems for abelian varieties over number fields]. Inventiones Mathematicae (in German) 73 (3): 349–366. doi:10.1007/BF01388432. MR 0718935.
Faltings, Gerd (1984). "Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern". Inventiones Mathematicae (in German) 75 (2): 381. doi:10.1007/BF01388572. MR 0732554.
Faltings, Gerd (1991). "Diophantine approximation on abelian varieties". Ann. of Math. 133 (3): 549–576. doi:10.2307/2944319. MR 1109353.
Faltings, Gerd (1994). "The general case of S. Lang's conjecture". In Cristante, Valentino; Messing, William. Barsotti Symposium in Algebraic Geometry. Papers from the symposium held in Abano Terme, June 24–27, 1991. Perspectives in Mathematics. San Diego, CA: Academic Press, Inc. ISBN 0-12-197270-4. MR 1307396.
Bombieri, Enrico (1990). "The Mordell conjecture revisited". Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (4): 615–640. MR 1093712.
Vojta, Paul (1991). "Siegel's theorem in the compact case". Ann. of Math. 133 (3): 509–548. doi:10.2307/2944318. MR 1109352.
Deligne, Pierre (1974), "La conjecture de Weil. I", Publications Mathématiques de l'IHÉS (43): 273–307, ISSN 1618-1913, MR 0340258
Deligne, Pierre (1980), "La conjecture de Weil. II", Publications Mathématiques de l'IHÉS (52): 137–252, ISSN 1618-1913, MR 601520
Bombieri, Gubler, Heights in Diophantine Geometry, http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511542879
Faltings, Wüstholz, Diophantine approximations on projective spaces, http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002111748