Technically: very little.
Kindle runs a version of Linux atop an ARM CPU just like the Pocketbooks. The SDKs are different but it wouldn't be too hard to port the reader app.
However, Kindle Reader relies on persistent connectivity for Whispernet and Amazon won't license it without Whispernet and the built-in storefront. Add in the economy-of-scale advantages of the Amazon hardware there really isn't much incentive for either party in the prospective deal.