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?

I'll start with a quote for context and to point to a helpful resource that might have an answer for the OP. It's from V. Amrhein, S. Greenland, and B. McShane. Scientists rise up against statistical significance. Nature, 567:305–307, 2019. https://doi.org/10.1038/d41586-019-00857-9

We must learn to embrace uncertainty.

I understand it to mean that there is no need to state that we reject a hypothesis, accept a hypothesis, or don't reject a hypothesis to explain what we've learned from a statistical analysis. The accept/reject language implies certainty; statistics is better at quantifying uncertainty.
Note: I assume the question refers to making a binary reject/accept choice dictated by the significance (P ≤ 0.05) or non-significance (P > 0.05) of a p-value P.
The simplest way to understand hypothesis testing (NHST) — at least for me — is to keep in mind that p-values are probabilities about the data (not about the null and alternative hypotheses): Large p-value means that the data is consistent with the null hypothesis, small p-value means that the data is inconsistent with the null hypothesis. NHST doesn't tell us what hypothesis to reject and/or accept so that we have 100% certainty in our decision: hypothesis testing doesn't prove anything٭. The reason is that a p-value is computed by assuming the null hypothesis is true [3].
So rather than wondering if, on calculating P ≤ 0.05, it's correct to declare that you "reject the null hypothesis" (technically correct) or "accept the alternative hypothesis" (technically incorrect), don't make a reject/don't reject determination but report what you've learned from the data: report the p-value or, better yet, your estimate of the quantity of interest and its standard error or confidence interval.
٭ Probability ≠ proof. For illustration, see this story about a small p-value at CERN leading scientists to announce they might have discovered a brand new force of nature: New physics at the Large Hadron Collider? Scientists are excited, but it’s too soon to be sure. Includes a bonus explanation of p-values.
References
[1] S. Goodman. A dirty dozen: Twelve p-value misconceptions. Seminars in Hematology, 45(3):135–140, 2008. https://doi.org/10.1053/j.seminhematol.2008.04.003
All twelve misconceptions are important to study, understand and avoid. But Misconception #12 is particularly relevant to this question: It's not the case that A scientific conclusion or treatment policy should be based on whether or not the P value is significant.
Steven Goodman explains: "This misconception (...) is equivalent to saying that the magnitude of effect is not relevant, that only evidence relevant to a scientific conclusion is in the experiment at hand, and that both beliefs and actions flow directly from the statistical results."
[2] Using p-values to test a hypothesis in Improving Your Statistical Inferences by Daniël Lakens.
This is my favorite explanation of p-values, their history, theory and misapplications. Has lots of examples from the social sciences.
[3] What is the meaning of p values and t values in statistical tests?
