Difference between "Better than" and "More than"

The difference is qualitative versus quantitative. When you refer to higher quality, you say better and when you mean a bigger quantity, you say more.
In a casual use, when detail really doesn't matter, and you only mean one over another in a general sense, you may use either word in most contexts.

In the given context, prefer takes over instead:
"Many users prefer the look and feel of A over B."

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.
In some 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.

"Better" refers to quality.
"More" refers to quantity.

It is seldom the case that you can have something be greater in quality and quantity referring to the same property. "Like" is one of those rare cases, where "I like this better" and "I like it more" mean the same thing. The confusion is due to your using of "prefer" in place of "like", where "prefer" does not have this same property of usage.

As other answers already pointed out, you would have to construct your sentences as "prefer X over Y" or "prefer X to Y", since you cannot "prefer X more than Y" ("prefer" already means exactly that, "to like something better than another").

It is also true that Gnome 2 was (and still is) better than Gnome shell. Shame the upgrade was forced on us.