Well, what if the proof is just an exhaustive search of the space? Say 'all even numbers between 4 and 2^10' can be written as the sum of two primes. Say the proof is just listing out of these numbers and the primes that compose them. The proof tells you it is true. But it doesn't tell you anything about a natural underlying truth that causes this. Do you not find something wanting in such a proof?
Many proofs are in some sense proofs by exhaustion. For example, to prove that no square of an integer ends in a digit '3', the typical proof I) makes the search space smaller by working modulo 10, and II) exhausts all possibilities.
In the end, it all boils down to aesthetics: going from infinitely many cases to a small finite number, as given above, is acceptable because the number is low, and because mathematicians are convinced the remainder is about as simple as it gets.
Going to about 2000, as in the four color theorem, is not, because 2000 is a lot, and, I think more so, because mathematicians aren't convinced that it is necessary to handle each of these cases individually.
And of course, mathematicians would agree that using 5 instead of 10 in step I above leads to a nicer proof. It requires less tedious work in step II and leads to a stronger result (squares never end in 3 or 8).
