Is visual inspection the only way to compare large datasets? I have two large data sets, in fact, one of them is even much larger than the other. 
Visually, there doesn't seem to be that much difference between them:

The actual data underlying the box plot isn't normally distributed and doesn't normalise well to transformations. They are roughly the same distribution (i.e. the YES and NO distributions for each algorithm), but the large data size differences make other tests a bit useless. I have applied the Two-sample Kolmogorov-Smirnov test, however this is probably wrong and it gives extremely significant results. 
My questions are:
1) Do statistical tests on large datasets produce significant results given even slight differences between the two samples? The 'slightness' being magnified given huge data points.
2) Is visual inspection better with large datasets rather than applying non-parametric and parametric tests in which certain underlying assumptions may be violated.
3) For this data, what is the best course of action?

Edit

My data has structure like :
My data is of the form: 
Name    Bind    miRNA
a       300     NO
b       500     YES
c       140     YES
d       2345    NO

 A: *

*Yes. This is one key problem with standard goodness-of-fit tests on large datasets.

*I would prefer visual inspection, as well as measures of effect size. Even if there is a large overlap in distributions, a 15% improvement in some KPI may be very useful. I wouldn't care too much about specific distributions, depending on your specific application. In addition, boxplots are rather crude ways of displaying data. Here are a few alternatives.

*Hard to say, since we don't know your data... My suggestion about effect sizes may already be helpful.
A: There is nothing wrong with statistical summaries of large data sets. If a method would be appropriate with N = 100 then it is appropriate with N = 100,000 or 100,000,000.
There is, however, something wrong with how most people interpret p-values. The answer to your first question is "yes", but that answer is just another indication that you should look at effect size, not p value.
Regarding your second question: Visual inspection is critical with large or small data sets. But numeric comparison is very useful with both, as well. Their utility is not a function of data set size. 
Regarding your third question: Use both visual and numeric comparisons. Choose both the visual and numeric method that 1) are suited to your data and 2) are suited to the questions you want to ask about your data.
A: I suggest summarizing the difference with a general robust measure that does not depend on normality: the concordance probability that comes from the Wilcoxon-Mann-Whitney two-sample test.  The concordance proportion estimates the probability that a randomly chosen value from group A exceeds a randomly chosen value from group B.  This can be generalized to your "pairing of pairs" setting in which you can estimate the probability that method 1 provides measurements that are "more concordant" between A and B than method 2 does.  This is implemented in the R Hmisc package rcorrp.cens function.
