Usually statistical tests are used to look for differences. But sometimes your goal is to prove that two sets of data are equivalent. A conclusion of "no statistically significant difference" is not enough to conclude that two treatments are equivalent. You've really need to rethink how the test is set up.
In most experimental situations, your goal is to show that one treatment is better than another. But in some situations, your goal is just the opposite -- to prove that one treatment is indistinguishable from another, that any difference is of no practical consequence. This can either be the entire goal of the study (for example to show that a new formulation of a drug works as well as the usual formulation) or it can just be the goal for analysis of a control experiment to prove that a system is working as expected, before moving on to asking the scientifically interesting questions.
Standard statistical tests cannot be used to test for equivalence.
A conclusion of “no statistically significant difference” between treatments, simply means that you don't have strong enough evidence to persuade you that the two treatments lead to different outcomes. That is not the same as saying that the two outcomes are equivalent.
A conclusion that the difference is “statistically significant” means you have strong evidence that the difference is not zero, but you don't know whether the difference is large enough to rule out the conclusion that the two treatments are functionally equivalent.
In any experiment, you expect to almost always see some difference in outcome when you apply two treatments. So the question is not whether the two treatments lead to exactly the same outcome. Rather, the question is whether the outcomes are close enough to be clinically or scientifically indistinguishable. How close is that? There is no way to answer that question generally. The answer depends on the scientific or clinical context of your experiment.
To ask questions about equivalence, you first have to define a range of treatment effects that you consider to be scientifically or clinically trivial. This is an important decision that must be made totally on scientific or clinical grounds.
Statistical methods have been developed for testing for equivalence. You can use either a confidence interval or a P value approach.