A mash-up allows you to succeed or to fail fast. If you fail you can rephrase your hypotheses at an early stage. Using the feedback you get you can build a better app and find out what it takes to build an app that people actually want.
It is also true that, by using APIs or SDKs of third parties you can rely on much larger platforms than yours and since it is proven technology it is less error prone. For example, if you want to integrate payments other than In App purchases you will of course use the existing solution of a payment provider.
You can take advantage of social networks, by providing a single sign on for your app. You could, for example, provide a way for the user to sign up or to login with his Facebook or login account. It lowers the registration barrier, resulting in higher sign up conversions. Not only does the user have to take fewer actions...