There are multiple benefits. I'll give you two very good reasons for building an app for your site:
– You can integrate your website/web experience into the Facebook chrome, enabling your users to use your service without leaving the Facebook environment at all, which some might prefer.
– You can enable your users to directly interact with other unknown Facebook users, if your service is for example a game, without having to apply your own technology.
– You get ranked among the other apps, which will give you the possibility of reaching a higher visibility for your service.
Benefits that come with the Facebook API as well as Apps:
– You can let your customers 'x' something on your website/in your web service, for example you can let them 'wow' a piece of clothing in your online shop, resulting in posts like 'Miranda wows the D&G summer dress on Shoplife'. This gives you much greater flexibility in expressing brand experiences and helps you drive the effects of your marketing efforts further.
– You can integrate automatic sharing into your website/service, meaning that users automatically share their activities on your site to their timeline, like reading an article or listening to a song or buying a book. Take a look at Spotify and the Huffington post for two excellent examples on how to use this function. This, among others, means that friends of your customers will automatically discover your site and its content. The effects on the virality of your content can be enormous.
There is a great ZDNet news article detailing some of the best practices for social news sites (scroll to the bottom of the article for convenience). Some of them are applicable for other purposes, too:
*Please excuse iPhone formatting.
See the answer on Quora: