If there are two different proofs for one theorem, at some level are the two proofs the same, or can they be fundamentally different?
In other words, if you have two proofs of a theorem, can one show that the two proofs are expressing the same thing in different ways, and then remove the redundancies and generate a "shorter" proof?
They can be "completely different".
For example, some existence results have both indirect proofs and constructive proofs. There is often no way to interpret the "indirect proof" as "essentially the same" as the constructive proof.
Or you have the many different proofs of Quadratic Reciprocity. Gauss's first proof, in the Disquisitiones Arithmeticae, is very constructive; it is done by recursion, and for example it shows exactly how to transform a solution of $x^2\equiv p\pmod{q}$ into a solution of $x^2\equiv q\pmod{p}$ when $p$ and $q$ are not both congruent to $3$ modulo $4$; whereas his third proof was purely combinatorial, counting certain objects, and his sixth used Gauss sums, again an essentially different approach. Eisenstein used infinite products for his fifth proof, Kummer used quadratic forms, Zolotarev used permutations; Auslander and Tolimieri used the Fourier transform, Weil used theta functions. These are truly essentially different approaches, with no easy way to pare them down to the same thing (unless you "pare them down" to the statement of Quadratic Reciprocity itself).
They can come as "completely different". Consider proofs in logic. A classical logician CL can prove some theorem T used a reductio type argument. A constructivist logician CO proving T proves T in a very different way. Were it the case that the proofs of CO and CL were fundamentally the same at some level, then (at least it seems so) that the proof theory of CO and CL should match exactly. But, of course, they don't, so the proofs of CO and CL are not fundamentally the same, and thus different. So, no to the second question also.
