  Motivated by recent experiments on Bi$_3$Mn$_4$O$_{12}$(NO$_3$), we study a
frustrated $J_1$-$J_2$ Heisenberg model on the two dimensional (2D) honeycomb
lattice. The classical $J_1$-$J_2$ Heisenberg model on the two dimensional (2D)
honeycomb lattice has N\'eel order for $J_2 < J_1/6$. For $J_2 > J_1/6$, it
exhibits a one-parameter family of degenerate incommensurate spin spiral ground
states where the spiral wave vector can point in any direction. Spin wave
fluctuations at leading order lift this accidental degeneracy in favor of
specific wave vectors, leading to spiral order by disorder. For spin $S=1/2$,
quantum fluctuations are, however, likely to be strong enough to melt the
spiral order parameter over a wide range of $J_2/J_1$. Over a part of this
range, we argue that the resulting state is a valence bond solid (VBS) with
staggered dimer order - this VBS is a nematic which breaks lattice rotational
symmetry. Our arguments are supported by comparing the spin wave energy with
the energy of the dimer solid obtained using a bond operator formalism. Turning
to the effect of thermal fluctuations on the spiral ordered state, any nonzero
temperature destroys the magnetic order, but the discrete rotational symmetry
of the lattice remains broken resulting in a thermal analogue of the nematic
VBS. We present arguments, supported by classical Monte Carlo simulations, that
this nematic transforms into the high temperature symmetric paramagnet via a
thermal phase transition which is in the universality class of the classical
3-state Potts (clock) model in 2D. We discuss the possible relevance of our
results for honeycomb magnets, such as Bi$_3$M$_4$O$_{12}$(NO$_3$) (with
M=Mn,V,Cr), and bilayer triangular lattice magnets.
