Why can't we accept the null hypothesis, but we can accept the alternative hypothesis? I understand it's reasonable only to not reject the null hypothesis. But why can we accept the alternative hypothesis?
What's the difference?

In addition to the answers given by highly experienced users here, I'd like to offer a less formal and hopefully more intuitive view.
Briefly, the "null hypothesis" is considered accepted, unless there is some compelling evidence to reject it in favour of an alternative.
It helps to look at it from the decision-making perspective. Tests---not only statistical---help us make decisions. Before performing the test, we have one course of action. After performing the test, we may either keep the course or change it, depending on the test result. The null hypothesis is the default course of action, given no or not enough information.
For example, imagine you are a flying an aeroplane. Without a reason to do otherwise, you'll probably fly it straight towards your destination. But, the whole time you'd be performing "tests", like checking your radar whether there is some unexpected obstacle on your path. If the radar shows no obstacle, you'll keep your course. This is the default decision, which you'd most likely make even if you had to fly without a radar. I mean, what else could you do? Wildly zigzag through the sky?
In this analogy, the null hypothesis is that there is no reason to change the course. You don't "accept" it as a result of the test, because it has already been accepted before you took a look at the radar. Only if you discover an obstacle, you'd reject it in favour of changing the course.
Or, as a more real-world example, imagine developing a new drug for a disease. The default status, before you perform any trials at all, is that the drug is not approved. You may run in vitro, in vivo, and clinical trials to prove that your drug is safe and helpful. If that fails, the drug remains "not approved". Again, there is nothing to "accept", or at least nothing with practical consequences. Only with compelling evidence of the drug's usefulness its status can change to "approved".
As you can see from the examples, which hypothesis is treated as "null" is somewhat subjective. For example, is "homeopathy works" null, or does it need evidence to be accepted? That depends on your prior beliefs and experience. If you grew up in a homeopathic home, you are likely to consider it to work by default and would't change your mind unless you see strong evidence against it (or maybe ever). But, this can get arbitrarily philosophical / psychological.
