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/
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/