What is the difference between a one-shot MPC protocol and a unbounded invocation of multiple MPC protocols?
Basically, there a many real-world problems where one might want to compute something on sensitive data several times.  Indeed, later in the paper you cite, the authors give several example, e.g.:


  "Medical Data: One may envision a huge database which contains the medical data of every patient in the US. To compute any global statistic on this data, one would not want to put complete trust in any single database. Instead, it is distributed to $n$ different databases. Each time they need to compute statistics on this data, they engage in an MPC protocol. As in the voting example, since these databases contain very sensitive information, an adversary may try to obtain this information via a leakage attack. Thus, to ensure security, the databases must run an MPC protocol that is secure
  against continual leakage."


The important words above are "each time".  It would not do much good to have a distributed database that was secure only the first time it was queried.