How do I compare the variance of a distribution of species in one area versus another?
These are just some preliminary ideas...

I think you should look at the seasonal distributions separately, since the bimodal distribution is the outcome of two fairly separate processes. The two distributions might also be controlled by different mechanisms, so that e.g. winter distributions might be more sensitive to yearly climate. If you want to look at population differences and reasons for these I think it is therefore more useful to study seasonal distributions separately. An exploratory analysis could be to compare distribution percentiles (north/south and east/west coordinates) to look at range edges, or to use a fixed number of edge observations to establish borders. The weighted centre-point of population distributions can be used to look for differences in overall position. If you have grid occurences, percentage overlap between species/populations could maybe also be useful.

If you haven't already, you look also look at Maxent, which is a widely used software for modelling species distributions and habitats. Look at Elith et al. (2011) for an overview. If you want to look at changes over time (even if this doesn't seem to be your main point) you should also check out dynamic occupancy models that use occurrence records to model species distribution while taking detectability into account, e.g. MacKenzie et al. (2009).

As for a simple test for differences in variance (basically a test of homoscedasticity), there is Levine's test, which is used to compare variances of distributions between groups. Bartlett's test is an alternative, but Levene's test is supposed to be more robust to non-normality. In R the Levene's and Bartlett's tests are found in library(car). However, these are only suitable for unimodal distributions so, again, you should look at seasons independently.