Why do we say "the same" and not "the different"?

Solution 1:

We say "the same [noun]" because the use of the word "same" indicates that the noun refers to something that has already been mentioned, or that we already know about—so we use the definite article before the noun phrase. "Same" can also be used with other determiners in appropriate circumstances, like "that same [noun]" (although it's hard to think of circumstances where it could be used with the indefinite article "a").

The word "different", as Yosef Baskin points out, does not indicate that we are talking about something already mentioned or already known. In fact, it suggests the opposite.

A word that can act either way is "other". Roughly speaking, when it refers to one of two things, both of which are known (or both of which can be inferred to exist), it is used with the definite article "the": "First I put on one shoe, then I put on the other [shoe]" (shoes are known to come in pairs). When it refers to "something else" that isn't already known to form a pair with the first thing mentioned, it is often used with the indefinite article, which is in most circumstances written together with the word: "Yesterday was a bad day, and it looks like today will be another."