What is the difference between failing to reject the null hypothesis and having evidence to support the alternative hypothesis?

Researchers rely on the p values that are generated for each of their statistical tests to determine significance. If the p value is larger than the alpha, then they are not different in a statistically significant manner, and therefore the values are not considered different. Consider these concepts in terms of the differences between null and alternative hypotheses, and discuss this question: What is the difference between failing to reject the null hypothesis and having evidence to support the alternative hypothesis?