Test of whole distribution for two samples I have a fundamental question: is it reasonable to use a statistical test to find out if 2 datasets are similar or not? Some comments have called out the T-test as being unable to answer that question. Why is the T-test insufficient? Furthermore, what other tests are there to determine differences in distributions for the two sample problem.
 A: 
I have a fundamental question: is it reasonable to use a statistical test to find out if 2 datasets are similar or not? 
Some comments have called out the T-test as being unable to answer that question. Why is the T-test insufficient? 

Two issues: 
1) A data set may have very similar mean to a second one, yet substantively differ from the other in other ways (different spread, different shape)
2) The usual hypothesis tests don't tell you if the data sets are necessarily "similar"; they may detect if they're more different than could be explained by random variation. You may be thinking of something more in the area of equivalence testing. Alternatively it may be that your underlying question is more closely related to looking at an effect size than hypothesis testing.

Furthermore, what other tests are there to determine differences in distributions for the two sample problem.

There are tests for many different aspects of distributions, it depends on what you're interested in. People may use a Levene test or perhaps Browne-Forsythe test to compare spreads, or if we look toward nonparametric test, a Siegel-Tukey perhaps an Ansari-Bradley test.
To compare the entire distribution, there are two-sample Kolmogorov-Smirnov tests.
[Note that hypothesis tests are really for making inferences about populations on the basis of samples.]
A: To be formal about it, there are very abstract non-parametric tests of whether or not the probability distributions of two samples are equal or not. A test that is powered to detect differences in samples under any conditions is the Kullback-Leibler test of divergence. It is a test of the two-sample strong null hypothesis,
$$\mathcal{H}_0: \mathcal{F}_0 = \mathcal{F}_1$$
Note there's no parameter here, like a mean. But what can we say about tests that are powered to detect an infinite number of differences under all possible conditions? They're very rarely practical. The mean is a useful summary statistic for virtually all probability distributions: uniform, beta, weibull, gamma, etc. If there is a useful difference between two distributions, usually there is a difference in means.
A: A t-test, like any hypothesis test, can show strong evidence that something is not like it should be. If one fails to prove than, one can't say that the opposite is true. The framework of hypothesis test consider some state as "the natural" state. This state is assumed if there are no evidence against it and cannot be proved. 
See Burden of Proof Fallacy
