What is the difference between an inertial instability and a symmetric instability?
Inertial instability is similar to the centrifugal instability in that we are looking at the stability of parcels to horizontal perturbations.  In the inertial case, however, the initial state is geostrophic balance rather than cyclostrophic balance.  

Symmetric instability is the case where a parcel is inertially stable to horizontal perturbations and statically stable to vertical perturbations but unstable to perturbations between the horizontal and vertical.  An additional constraint is that the flow is symmetric and only varies in two dimensions.  This instability was also first examined as centrifugal instability, but with added baroclinity.  Conditional symmetric instability (CSI) is related to symmetric instability with the difference being that $\overline{\theta}^*_e$ rather than $\overline{\theta}$ surfaces are used to asses the instability. 

For more information on these mesoscale instabilities, you can find information on them in Markowski and Richardson (2010) pages 49 (inertial instability) and 53 (symmetric instability).  Holton (4th ed, 2004) treats these instabilities on pages 204 (inertial oscillations) and 279 (symmetric instability).

Markwoski, P., and Y. Richardson, 2010: Mesoscale meterology in midlatitudes. Wiley, 430 pp.
Holton, J. R., 2004: An introduction to dynamic meteorology. Elsevier, 4th ed., 535 pp.