Difference between MPC and outsourced computation
About outsourced computing: under a security viewpoint, there is no challenge in trusting a thirty party and give him all responsibility on doing the task in secrecy and return the output the the parties. Trust a central party is all everyone doesn't want. So security isn't a problem when you trust someone. 

Secure multi-party computation: in the most classical view, with secure MPC we have n parties entering their correspondent inputs (in secrecy), and get their correspondent output(s) (when it is the case) (in secrecy). The challenge here is the inexistence of a trusted party, and all we want is that a protocol makes the role of that trusted party, giving the guarantees that trusted party could provide.  

For example, we can consider some security level of a digital version of an election: think about some computer that we can trust, in a hypothetical idealized world, realizing the election function. Your challenge is: how can you construct a (distributed) protocol that runs among the parties, and that gives all the security guarantees that trusted computer could provide to the electors.