The technical difference, as far as I'm aware, is that the Amazon version works by using Javascript on your browser to identify the content block and send it to your account. We make a separate request for the content from the server. The JS method works better in many cases (e.g. if the page you're viewing is not publicly accessible - paywall perhaps). The implementation, however, is problematic for many people.
See user reviews of the Amazon version at https://chrome.google.com/webstore/detail/send-to-kindle-for... and https://addons.mozilla.org/en-US/firefox/addon/sendtokindle/
User reviews of our extension: https://chrome.google.com/webstore/detail/push-to-kindle/pna... and https://addons.mozilla.org/en-US/firefox/addon/kindle-it/ and https://play.google.com/store/apps/details?id=org.fivefilter...
Another difference is that we offer an Android app and an email service to get articles to your Kindle account. Amazon does not offer this themselves yet. Their send to Kindle offering on Android is for personal documents on the device itself.
I should also add that when we released Push to Kindle, Amazon did not offer their own browser extensions. So although we weren't the first, we weren't competing with Amazon.
That said, this looks pretty clean.
Ref 1: http://www.theguardian.com/money/2012/oct/22/amazon-wipes-cu...
Ref 2: http://boingboing.net/2009/07/30/high-school-student-1.html