Can complexities differ w.r.t. different computational models?
Niel De Beaudrap's point is an important one: a complexity class is defined with respect to a machine model. But if I were to re-interpret your question as:


  Can the complexity of a problem differ greatly in different computational models ? 


Then the answer is yes. JeffE's answer to my earlier question about decision tree complexity presents an example of an NP-hard problem (SUBSET SUM) that has polynomial decision tree complexity.