Bioweapons do not have similar dual-use beneficial purpose as the AI does. As a result, AI development will continue regardless. It can give competitive advantage on any field.
Bioweapons are not exactly secret as well. Most of the methods to develop such things are open science. The restricting factor is that you potentially kill your own people as well, and the use-case is really just a weapon for some mad man, without other benefits.
Edit: To add, science behind "bioweapons" (or genetic modification of viruses/bacteria) are public exactly for the reason, that we could prevent the next future pandemic.
The most risky parts of biotech, the ones directly related to bioweapons, are not made publicly accessible - but it's hard, as unlike with nukes, biotech is dual-use to the very end, so we have to balance prevention and defense with ease of creating deadly pathogens.