If you gather requirements, you are assuming the business people or users know what the requirements are and can give them to you - that is, they can express the requirements for their system or enhancement clearly and concisely. Gathering implies that something is already there to gather. It is a rare circumstance when the users or stakeholders or anyone in the business has a clear understanding of the detailed requirements that will solve their problem. It's hard enough to find business people who can even state their problem concisely and clearly.
Eliciting is gathering information with understanding. The business analyst asks questions and more questions and analyzes the information gathered to produce the requirements - what is necessary to solve the problem. Investigation might be a better word for what business analysts do.
You gather apples or wheat.
You elicit information which you then analyze into requirements. That is the business analyst's forte. We are after all, analysts, not gatherers.