From a descriptive standpoint, a QQ plot is about as good as it gets for something like this, you can really see where in the distributions things are different (heavier tails, skew, location shift, outliers). That might give you insight about what parameter of the distributions you are interested in testing.
The issue with a statistical test is that it demands that you really clearly state what you care about, i.e., your null hypothesis. As an example, if you wanted to compare whether or not the mean of two distributions was different, you might not care that their standard deviations are different:
a = rnorm(100, sd = 3); b = rnorm(100, sd = 1))
> t.test(a,b)$p.val
[1] 0.2825168
> ks.test(a,b)$p.val
[1] 7.14257e-05
If our null hypothesis is that the means are different, it wouldn't be correct to test with the KS-test and we see we'd incorrectly reject the null. In your case, you have to state, what aspect of the distribution you care about and then test accordingly.
It would be fine to ask, Is the 10th-percentile different -- that's a good null hypothesis. I'm not aware of well-known statistical test for such a thing, but you could certainly use a bootstrap-based approach.
Best Answer
There might not be a best solution as such. But if you are trying to visualize the distribution similarity, you can use violin plots instead of box-plots.
However, if statistical measures of similarity is what you are trying to calculate, then I think you should try fitting the distributions against several standard distributions and see how well they compare. If you are familiar with GLDs (Generalized Lambda Distributions), you can obtain GLD fits for each of the distributions and compare their resulting parameters.