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?

Within the Bayesian framework you can "accept the null hypothesis" in the sense that the posterior probability of a point null hypothesis can tend to one with increasing sample size.  This requires that the null hypothesis is exactly true and that you're willing to represent this in your prior by a point mass.  Lindley (1957, p. 188) gives two examples where this is arguably reasonable: testing for linkage in genetics, and testing someone for telepathic powers.  In addition, your prior on the parameter of interest must be proper under the alternative hypothesis. See for example this answer.
