Proofs that expose a deeper structure The standard proof of the Chernoff bound (from the Randomized Algorithms textbook) uses the Markov inequality and moment generating functions, with a bit of a Taylor expansion thrown in. Nothing too difficult, but somewhat mechanical. 

But there are other Chernoff bound proofs that expose the deeper structure driving the result. For example, there's an information-theoretic version that goes via the method of types, exemplified by this paper of Impagliazzo and Kabanets, as well as this brief post by Sanjoy Dasgupta. These latter proofs are more "intuitive" in that they provides a generalization of the standard result, as well explaining where the funny terms in the exponent come from (it's a KL-divergence). 


  What are good examples of such things
  ? To be more concrete, here are the
  rules:
  
  
  The statement should be reasonably well-known (the kind of thing that
  would be taught in some kind of
  graduate class)
  There should be a "standard" proof available in textbooks or standard
  reference material that is "commonly"
  taught
  There should be an alternate proof that is not so well known, is NOT
  commonly taught, and either proves a
  more general statement or links the
  statement to a deeper mathematical
  structure.
  


I'll start off with two examples. 


The chernoff bound


"textbook" proof: markov inequality, moment generating functions, Taylor expansion (MR)
Uncommon and insightful proof: method of types, exponent of tail involving KL-divergence

The Schwartz-Zippel Lemma


"textbook" proof: base-case involving univariate polynomial. Induction on number of variables
"uncommon" proof: geometric argument via Dana Moshkovitz (and Per Vognsen)



One example per answer please. 

p.s I'm not necessarily implying that the uncommon proof should be taught: a direct proof is often easier for students. But in the sense that "proofs help us understand", these alternate proofs are very helpful.
I'm not sure this is quite what you're looking for, since I've seen the "uncommon" proof in textbooks, but: the O(n log n) time bound for quicksort.


"Textbook" proof: set up a randomized recurrence relation, prove by induction that it has the desired solution.
"Uncommon" proof: find a simple formula for the probability that any two elements are compared (it's just 2/(d+1) where d is the difference between their ranks in the sorted order), and use linearity of expectation and harmonic series to compute the expected number of pairs that get compared.


The textbook proof requires less creative insight, but the uncommon proof introduces a technique that's very useful in other algorithm analysis e.g. for randomized incremental algorithms in computational geometry.