You can use either word, although I would go for "different", myself. "Another" has a slight nuance of "one more", so it may be better used to describe the new one compared to the old one, rather than the reverse. I have no problem with starting a sentence with "but"; this is one of these bogus rules which have no basis in reality, like the one one about "none" taking only a singular verb, or using "which" as a restrictive relative pronoun (as I just did). To see what people at the Oxford Dictionary have to say about this, see http://blog.oxforddictionaries.com/2012/01/can-i-start-a-sentence-with-a-conjunction/.

To me both words work and are interchangeable in this sentence, as long as you remove "a" before using "another"

Do not start a sentence with "but". Last year I attended a DIFFERENT school. But then I left for this one sounds better to me.

"Last year I attended a DIFFERENT school, but then I left for this one" I would say that 'different' is the better choice. (Try not to start a sentence with 'but'.) Some families have to move house often because of work location, so their children might say: "Last year I attended a DIFFERENT school, but I had to go to another school after we moved."