I am not sure that "better" really is preferred to "more" in sentences like yours. "More," as well as "better," is not unnatural, and is perfectly grammatical
With the verb "like," you can use "better" to mean "in a more excellent way," as in this entry for "better" from the American Heritage Dictionary*:
adv. Comparative of well2.
"¢ In a more excellent way.
A second listing gives this definition:
"¢ To a greater extent or degree better suited to the job; likes it better without sauce.
So, "which do you like better" in your sentence can mean "which do you like in a more excellent way – summer or winter," or, "which do you like to a greater degree or extent – summer or winter"?
Better" also goes with "like" in this entry for "prefer" in the COBUILD**:
"¢ If you prefer someone or something, you like that person or thingbetter than another....
However, logically, with "like" you could use "more" as the comparative of "much," as in this entry from the American Heritage (substituting "like" for "love"):
adv. Comparative of much.
a. To or in a greater extent or degree: loved him even more.
Google shows 22,400 examples of "like better than," in examples like these:
"¢ Poetry I Like Better Than James Tate
"¢ I'll never be anywhere I like better than the veranda off my childhood bedroom in Palo Alto, California.
"¢ Or vote for all the candidates whom you like better than what you expect from the election
and 76,200 for "like more than" in examples like these:
"¢ LikeMore than Cost Accounting
"¢ Five things I like more than I want to admit.
"¢ Links to sites we like more than ours
Each of the examples of "like better than" could be exchanged with "like more than," and vice versa.
I would tell my students that "like better than" and "like more than" are both correct in many sentences. If you want to focus on how excellent something is, how it causes you to like that something more than something else because of its excellence, "like better than" is very appropriate.
If you want to focus on the quantity of feeling with which you like something, "like more than" is very appropriate.
I would not be strict in preferring one adverb over the other. In many cases, "like better than" and "like more than" are interchangeable, and both are correct.
As a note, which adverb to use might be a stimulating discussion, like this:
Teacher: Which team are you rooting for, Jim?
Jim: Oh, the Panthers of course. I like the Panthers better than any other team
Tom: I like the Tigers more than the Panthers.
Teacher: You do? Why?
Jim: The Panthers play so well all the time. I like them because they are excellent. The way I like them is a special way – it's almost a love for them that I have.
Teacher: Jim, what about you?
Tom: Well, the Tigers are always good, too. I like them a whole lot. I like them more than I like any other team. I have a great quantity of love in my heart for them.
Or, any such foolish dialog that might highlight the small difference between "like better than" or "like more than.' Actually, you might not come to a conclusion of which phrase is better, since often there is virtually no differrence.
*The American Heritage Dictionary of the English Language. Fourth Edition. Harper Collins. 2003
**The Collins COBUILD English Dictionary. Harper Collins.. 1995