Are p-values and critical values two different hypothesis testing methods?

They are two ways to reach the same decision. With the critical value approach, compare the test statistic to the rejection cutoff. With the p-value approach, compare the p-value to the significance level.

For a 5% test, a p-value of 2% means the result is statistically significant at 5%, so reject the null. A p-value of 8% means the evidence is not strong enough at 5%, so fail to reject. The critical value approach should give the same reject or fail-to-reject conclusion when the test is set up correctly.