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?

The answer depends on whether you are using a pre-defined critical value (or p-value threshold like p<0.05) in a hypothesis test that yields a decision (a Neyman–Pearsonian hypothesis test), or whether you are using the magnitude of the actual p-value as an index of the evidence in the data (a [neo-]Fisherian significance test).
If you are doing a hypothesis test then you are working with a set of rules that grant you a pre-set confidence of long-run performance of the test procedure. The way that the rules can give confidence about long-run test performance is by specifying what decision applies depending on the data, and the decision relates to the acceptance or non-acceptance (yes, that is rejection as far as I am concerned) of the null hypothesis. Rejection of the statistical null hypothesis can be thought of as acceptance of another hypothesis, but that other hypothesis can be nothing more than a set of all not-the-null hypotheses that exist within the statistical model. Accepting that not-the-null hypothesis is not very informative and so it is not unreasonable to simply say that the test rejects the null but does not accept anything else.
There is (sometimes) a specific 'alternative' hypothesis specified for a hypothesis test: the hypothetical effect size plugged into the pre-experiment power analysis used to set the sample size. That 'alternative' hypothesis IS NOT tested by the hypothesis test and has very little meaning once the data are available.
If you are doing a significance test then a small p-value implies that the data are inconsistent with the statistical model's expectations regarding probable observations where the null hypothesis is true. The analyst can then use that evidence to make a scientific inference. The scientific inference might well include an interim rejection of the statistical null hypothesis and acceptance of a specific 'alternative' hypothesis of scientific interest. It depends on the information available and the scientific objectives, and it is a process that is very rarely considered in statistical instruction.
See this open access chapter for much more detail: https://link.springer.com/chapter/10.1007/164_2019_286
