I just learned (thanks to Harry Gindi's answer on MO and to Qiaochu Yuan's blog post on AoPS) that the chinese remainder theorem and Lagrange interpolation are really just two instances of the same thing. Similarly the method of partial fractions can be applied to rationals rather than polynomials. I find that seeing a method applied in different contexts, or just learning a connection that wasn't apparent helps me appreciate a deeper understanding of both.
So I ask, can you help me find more examples of this? Especially ones which you personally found inspiring.