Translation:The dress that I like more is that black one.
In this case, it is used as a pronoun, and when it is used as a pronoun it only becomes questo / questa / questi / queste
quello = that one (when used as a pronoun)
While "questo" is pretty easy - it is the same as a demonstrative and as a pronoun - quel is a bit different.
"quel" as a demonstrative takes on forms depending on gender and first letter of the following word (for masculine words that would be "quel, quello / quei, quelli"). Used as a pronoun it is much simpler and very similar to "questo" - "quello / quelli" for masculine and "quella / quelle" for feminine.
Technically, no, not in this context. "That" is a restrictive pronoun. Here's one explanation: https://owl.english.purdue.edu/owl/resource/645/01/
I too wrote "the dress that I prefer is that black one", but unlike wulfrunian I didn't think of it as being clever but merely how I would translate it to someone who didn't know any Italian. The better translation of these two is the shorter equivalent; "prefer" is one word, "like more" is two words and clumsier. I prefer "prefer".
Why is "prefer" marked wrong? Neither of you who asked "why not 'prefer?'" admitted reporting "my answer should be accepted".
If you don't do that, then this list will continue to grow without bound as new unsuspecting students fall for the trap of using "prefer" only to find that DL has not yet discovered "prefer" to be equivalent in meaning to "like more".
It just doesn't mean exactly the same thing. The sentence is saying that "Of all the dresses in front of me (or that I own or whatever), I like that black one best." Saying "I like that black dress more" removes the idea of there being a distinct set of options to choose from, instead implying a single other option. It also goes from 'most' to simply 'more'.
Think about it as "mi piace" = "pleases me". I wonder why this isn't accepted as a translation, though -- isn't it idiomatic enough in English? For a thorough explanation see http://blogs.transparent.com/italian/a-different-point-of-view/