A person familiar with Google's thinking on the matter told Engadget that a big point of contention had been the fact that Amazon implemented what was essentially a hacked version of YouTube on both the Echo Show and Fire TV. Rather than work with Google to build versions of its apps that work on Amazon's devices, Amazon has been trying to do it itself -- a move that cuts out features and also likely affects Google's ability to collect on some of the ad revenue that comes from its videos.
We've learned that Google is pulling support for YouTube from the Echo Show as of today; the version of YouTube that exists on Fire TV will work until January 1st.
Google and Amazon have been in discussions for a long time to try and have a more mutually beneficial relationship, but it sounds like it hasn't been going well. Our source says that Google would like to have more of its hardware products like Chromecast and Google Home available through Amazon, but the company has long avoided selling products that compete with its own hardware initiatives -- that's why you can't get an Apple TV on Amazon, for example. Similarly, Amazon Prime Video doesn't support Google's Cast feature, making it difficult to get video to a big screen for Android users.
Between those difficulties and the fact that Amazon pulled some of Nest's recent products (including the new Thermostat E, Camera IQ and Nest Secure alarm system), Google doesn't seem inclined to let Amazon continue to run its "hacked" versions of YouTube on its devices, though our source says that Google would be much happier working directly with Amazon than forcing consumers to pick sides.
That's not an unreasonable position to take, as the company surely wants more control over the YouTube experience on Amazon's products, but unfortunately the only people getting hurt in this feud are customers. Google seems to be betting that Amazon will want to get YouTube back up and running on its flagship video products before long, so it can afford to take a hard line here. Whether or not things get resolved by the end of the year remains to be seen, though.
Update: Amazon has responded, and its response points to the unfair nature of Google effectively blocking access to the YouTube web site based on the type of device being used to access it. "Echo Show and Fire TV now display a standard web view of YouTube.com and point customers directly to YouTube's existing website," Amazon's statement reads. "Google is setting a disappointing precedent by selectively blocking customer access to an open website. We hope to resolve this with Google as soon as possible."