How do POMDPs and Dynamic Influence Diagrams differ?
I'll quote from the paper "Interactive Dynamic influence diagrams" (DIDs) by Polich and Gmytrasiewicz (AAMAS 2007):


  A dynamic influence diagram is a computational representation of a POMDP. 


They continue soon afterward:


  DIDs perform planning using a forward exploration technique known as reachability analysis. This technique explores the possible states of belief an agent may be in in the future, the likelihood of reaching each state of belief, and the expected utility of each belief state. The agent then adopts the plan which maximizes the expected utility. DIDs provide exact solutions for finite horizon POMDP problems, and finite look-ahead approximations for
  POMDPs of infinite horizon.


To me this suggests that DIDs are, as the authors state, one possible representation of a POMDP, to which a specific look-ahead type solution method is related to. As look-ahead is limited to approximate solutions for infinite horizon POMDPs, this also suggests that DIDs can not represent every POMDP.