Here is a list of problems that may or may not qualify as "sufficiently" different.  By the same proof as for Graph Isomorphism, if any of them is NP-complete, then the Polynomial Hierarchy collapses to the second level.  I do not think there is any broad consensus as to which of these "ought" to be in P.

 - Graph Automorphism (determine if a graph has a nontrivial automorphism).  Reduces to Graph Isomorphism, but not known (not thought?) to be GI-hard.
 - Group Isomorphism and Automorphism (where the groups are given by their multiplication tables).  Again, reduces to Graph Isomorphism, but not thought to be GI-hard.
 - Ring Isomorphism and Automorphism.  In a sense, this is the grand-daddy of all the above problems, since integer factoring is equivalent to finding a nontrivial automorphism of a ring, and Graph Isomorphism reduces to Ring Isomorphism.  See <a href="http://dimacs.rutgers.edu/~kayaln/pdf/ringAuto_journal.pdf">Neeraj Kayal, Nitin Saxena. Complexity of Ring Morphism Problems.
Computational Complexity 15(4): 342-390 (2006).</a>  (Interestingly, _determining_ if a ring has a nontrivial automorphism is in $P$.)
 - <a href="http://blog.computationalcomplexity.org/2010/07/what-is-complexity-of-these-problems.html">This post</a> by Bill Gasarch contains some other problems with the taste of Ramsey theory that look like they might be intermediate.
 - By Mahaney's Theorem, no sparse set can be NP-complete.  But we also know that there are sparse sets in $NP$ - $P$ iff $NEXP$ is not equal to $EXP$.  So assuming $NEXP \neq EXP$, the padded version of any $NEXP$-complete problem is of intermediate complexity.  (Such a set cannot be in $P$ unless $NEXP = EXP$, contradicting our assumption.)  There are plenty of natural $NEXP$-complete problems.