neither nor comma

1) 'Neither' and 'Nor' are a pair of correlative conjunctions (other examples, either...or, not only...but also). When pairs of such conjunctions are used in a sentence, a comma to separate them is not required. Hence, your sentence without a comma is correct. Note that there is an exception to this rule. For example, if commas are required to offset a nonrestrictive clause, then they must be retained. For example: Neither John, who was ill, nor Oliver attended the party.

2) The word 'Neither' indicates that two ideas are linked together in a sentence. Hence, to link the two ideas in your sentence (you don't like broccoli and you also don't like tomatoes) comma must be removed.

Hope I was able to explain it in a simplified way. Best Regards!