Q:

Why are p-values probabilities rather than likelihoods?

The p-value is the probability of obtaining test results at least as extreme as the results actually observed during the test, assuming that the null hypothesis is correct.
Why is the p-value a probability rather than a likelihood function? 

A:

Likelihood applies to data once observed. If you were describing a property of this data set, you would be speaking of a likelihood. Probability describes a property of hypothetical data sets not yet observed. Against the arbitrary standard of this data set: the probability of observing another data set which departs from the  null hypothesis as much as or more than this data set does.

