'alexa-app' is an NodeJS library to simplify development of Alexa Skills. This is an awesome alternative to the Amazon's 'alexa-sdk'. Why would you need an alternative? If you want to deploy your app on non-amazon products like Heroku. It also supports a Finite State Machine which makes it easier to develop flow based apps and avoid redundant code related to an intent as opposed to the 'alexa-sdk'. Alexa-app also has some awesome developer tools like chatskills which make development a breeze. Depending on your requirement and application, both alexa-app alexa-sdk, can serve the purporse.
@marcelobern highlights some more differences between them if you want to decide which one to choose. https://goo.gl/SPkFlP
