What user info does Apple disclose to third party app developers, once a user purchases an app?
The short answer is it depends.
It depends on what the App developer claims it needs to collect. Apple doesn't sell your data nor will it give out data "willy nilly." But if a product/service that you want requires the sharing of data, Apple will share it.
From Apple's Legal page concerning Privacy:
Disclosure to Third Parties
At times Apple may provide third parties with certain personal information to provide or improve our products and services, including to deliver products at your request, or to help Apple market to consumers. When we do, we require those third parties to handle it in accordance with relevant laws. Apple does not sell personal information, and personal information will never be shared with third parties for their marketing purposes. For example, when you purchase and activate your iPhone, you authorize Apple and your carrier to exchange the information you provide during the activation process to carry out service, including information about your device. If you are approved for service, your account will be governed by Apple and your carrier’s respective privacy policies.
As for your data with respect to individual apps, what gets shared depends on what the App itself is requesting. However, it's Apple's policy that you be made aware of what personal info the app requires
Before purchasing/installing an app, check to see what info the app collects. Many people marvel at how many free apps there are on the App store not realizing the hidden cost - their privacy. App developers want to get paid for their work and if they're not charging you anything up front or have in-app purchases, the app isn't the product they're selling - you are.