Topic targeting is a broad way of targeting people who've expressed interest in that topic or "related topic". It's a good option if you want to create awareness, do branding or generate likes. But when your end goal is sales/leads - it's good to go for precise interests as that's like exact match - and it ensures that you're reaching only those people which you're actually targeting - not related ones.
Precise interests are more specific and tend to perform better. The broad categories tend to be too general. They get low CTRs leading to poor performance in direct response campaigns. You're better off bundling related, precise interests together within an ad/campaign.
Topics: Broader reach. With topics you can target all users who like any variation of a word or phrase. So if you want to target people who like Kraft Cheese, use #cheese and have the ability to target users who like unofficial pages like “I ♥ Cheese”.
Precise Interest: What it sounds like - more precise. This is the original way to target fans of a specific page. The newer Broad Category Targeting is a way to advertise to folks with similar profile characteristics. For example, soccer, newly engaged, etc.
Hope this helps! Let me know if you have any follow-up questions. I've been using Facebook ads for years :)