Every where, p-value is used to understand the significance of a hypothesis.
I will refer to 2 programs which refer to p-value for reliability / statistical significance.
The question we are asking is:
The biological way: How similar are sequence ?
How possible is a chance that the target sequence is a randomly shuffled nucleotide string of the reference sequence?
The p-value gives a value which indicates a numerical value to this question. We are not interested in knowing the similarity between the target sequence and a random string of nucleotides. Hence the lower the significance to this hypothesis, the higher is the confidence that the two sequences under comparison are statistically different and not by chance or random aggregation of nucleotides.
(correct me if this is not the case)
The obvious question we want to ask is:
How different the expression values are between the experiment and the control (biologically)?
The hypothesis question is: How possible is a chance that the difference between the two values of expression is by coincidence?
So, the lesser the value about the possibility of coincidence, the more significant and reliable the expression and hence the differential expression.
(correct me if this is not the case, the question can be put in a better also I suppose).
Enjoy.