Is there a statistical test comparing two distributions? So I have a bar chart with x-axis: microRNA base additions (+1, +2, +3) and y-axis: microRNA counts of patients (each dot is a patient).

I would like to compare distribution of base additions for Tumour vs. Normal. 
I was thinking of Chi square test, but I was not sure if I should calculate means of microRNA counts of patients or take medians? 
Is there any statistical tests in R where I can compare differences in the distribution of base changes in Tumour vs. Normal? 
 A: One classic way to deal with count data that takes multiple factors into account is Poisson regression. That's a type of generalized linear model. In a way you've already started down that path by looking at the logarithms of the counts; that's the standard link function for Poisson regression. Poisson regression can be considered an extension of chi-square categorical analysis for more complicated designs.
Poisson regression is based on a model of how the number of counts in an observation are determined. So think about that first. Each of your data points is the sum of the counts of microRNAs that have been altered in size as indicated along the x-axis of your plot. The number of counts seems to depend on the magnitude of the alteration, whether the sample is tumor or normal, and the individual from whom the samples were taken.
The data indicate that there is not a simple tumor-normal difference that is constant in terms of counts; tumors seem to have higher counts than normals with positive 3' end modifications, but lower counts than normals for negative 3' modifications. So you will need to take into account an interaction term between tumor/normal status and the magnitude of the 3' end modification.
Then you probably should take into account the influence of the patient on the number of counts in an observation. Do patients simply differ in terms of the total numbers of counts as a function of tumor/normal status and 3' end modification length? If so, then with what seems to be about 30 patients a simple random effect accounting for baseline differences among them would be appropriate. That would mean you have a mixed model, incorporating both fixed effects (tumor/normal, 3' end modification length, and their interactions) and random effects (patients).
The glmer() function in the R lme4 package provides for generalized linear mixed models. What I just outlined would be expressed as:
glmer(counts ~ tumorStatus * endChangeLength + (1|patient), data = yourData, family = poisson)

where the data are structured with 1 row per data point shown on your plot and columns indicating the number of counts, the tumor/normal status as a binary predictor, endChangeLength as a 7-level categorical predictor (representing which of the 3' end change lengths the counts were obtained from), and an ID for the patient.
Rather than just plow ahead with this, however, take some time to think about your data in more detail and enlist someone locally with statistical expertise to help. Mixed models can be powerful but they can be tricky in several respects. Also, there might be details in your experimental design and in the goals of your study that can't be adequately explained in a question on a site like this. Such details might affect how you structure your model, whether you need to consider other random effects for individuals, and whether you need to take into account the identities of the modified microRNAs as well as their overall sums of counts. Explaining just what you wish to accomplish to someone in person can greatly advance your work.
A: Maybe you can try ks.test
> ks.test(rnorm(50),rnorm(30))

    Two-sample Kolmogorov-Smirnov test

data:  rnorm(50) and rnorm(30)
D = 0.18, p-value = 0.5272
alternative hypothesis: two-sided

> ks.test(rnorm(50),runif(30))

    Two-sample Kolmogorov-Smirnov test

data:  rnorm(50) and runif(30)
D = 0.56, p-value = 6.303e-06
alternative hypothesis: two-sided

