Consumed service: Your app sends a message to another app and that app sends a response
Published service: Another app sends a message to your app and you send response
App service - Service works only between Mendix apps
Web service - works via REST or SOAP and works with any other service that supports REST/SOAP
In short, published means that you expose an app service or a webservice. Consumed is the other way around, your Mendix app uses somebody else his app or webservice.
Now the difference between an app and a webservice. Webservices are generic. Appservices is when you communicate between two Mendix apps. Then you have a lot of more option then with a regular webservice.