Why might a planar, conjugated molecule be higher in energy than a nonplanar system?
I reproduced your calculations with PBE/def2-SVP (using Orca). The energy "jump" is even smaller than with MP2, however the causes are probably the same. So here is my answer:


The planar structure is a Transition State, with one negative frequency.
The dihedral you are scanning is probably not the best to describe this twisting.


Now in more detail. Here you can see the Minimum Energy Path (calculated with pDynamo using the NEB method) between the two symmetrically related minima.
 You can see that, as expected, the path is smooth.

The main change between the structures is the pyramidalisation of the terminal NH$_2$. This can be described by the N-C-N-H dihedral, which also changes smoothly as you can see here:


I don't think the reason why the terminal NH$_2$ is not planar is steric problems as @Ivan Neretin suggests. I think its more a question of nitrogen hybridisation, which is between sp$^2$ and sp$^3$. It is the same reason why aniline is not planar. This is discussed in these two questions:


Is nitrogen of aniline sp&#178; or sp&#179; hybridized?
How to rationalise the resonance structures and hybridisation of the nitrogen in a conjugated amine?


Hope this helps!