As far as I understand, word-choice and word-request are both about choosing a word, or a phrase.
The first could be used when the OP is asking about choosing between two or more words, while the last could be used when the OP is not restricting the possibilities. Still, also the questions using word-request could be understood as questions about choosing a word, where the choice is limited to the full dictionary of English words. For example, Is there any other neutral word for homosexual male than gay? could be interpreted as "What neutral word should I choose to mean an homosexual male?"
Is there any difference between the tags?