"I do not like vegetables."
I know they are different. Grammatically at least. Cant explain it easily... sukijanai is like saying something is unlikable. When instead you change the copula to its negated form, its like saying is not vegetables that are likeable. I wonder if the de wa version leaves open the possibility that you like other vegetables? Or it indeed makes it sound more rash or awkward Idk would love to hear from a native...
It's the fact that Japanese doesn't translate to English one-to-one, and translations often have to resort to giving you an equivalent rather than the exact words.
Here, for example, there is no 'I' in the Japanese sentence, and yasai is actually the subject. It literally translates to "(the) vegetables aren't liked", which is equal to "(I) don't like (the) vegetables".
は=topic. "Regarding x, ..." が=subject. From above + other readings: 好き always needs が, except in a negative sentence.
I remember it by thinking of it as a politeness, to not be direct (が) when saying "I don't like something".
For more authoritative answers, look up "wa vs ga japanese". Here is one: http://nihonshock.com/2010/02/particles-the-difference-between-wa-and-ga/
Its funny to see that "肉 は 好き じゃない" which is the same sentence but say meat instead of vegetables, have a lot of comments about how people actually loves meat, while here people are just asking for the correct anwer or pronunciation and stuff like that. its like, people actually hate vegetables.