In terms of its impact on methodology, the most important difference between a hypothesis test and a confidence interval estimation is this:
When conducting a hypothesis test, a claim has been made about the value of a population parameter. This claim becomes a hypothesis for the test, the null hypothesis. In other words, we assume that the claim is true and that the population parameter does equal the value assigned to it in the null hypothesis.
Let's say that again: when we conduct a hypothesis test, we assume that the null hypothesis is true. We maintain this assumption right up until the end of the test, and we work under this assumption. Indeed, we keep working under this assumption unless we find evidence not to. This is an integral part of the methodology of hypothesis testing.