Your insertion of the word "always" in your question is bound to generate some dissent.
There are plenty of uses of the terms more than and better than where the two would not be interchangeable. Consider:
On the violin, Roger performs better than Felix.
When it comes to keyboard
layouts, some purists believe DVORAK is better than QWERTY.
Manchester played better than Liverpool last night, but Liverpool
still won the game.
Such a nice evening! The weather doesn't get
any better than this.
That garage band is sounding much better
than they did last year – thank goodness!
Listen, Robert, I've had more than enough of your shenanigans!
The cost of gasoline is more than it was last year.
Olympic events, strength matters more than speed.
Some rules of
geometry get very complicated in more than three dimensions.
More than anything else, communication is the key to a happy marriage.
Clearly, the use of more than and better than are not interchangeable in those examples. This is unsurprising, as more is not always the same as better. (At the dinner table, for example, more food and better food are two very different things).
Still, there are some instances where either one could be used, particularly when we are talking about preferences:
Most children like vanilla ice cream more than chocolate.
No! I disagree! Most kids like chocolate better than vanilla.