404 with post name permalinks

I was able to figure out the problem, and for the sake of posterity, I’m posting the answer here. Basically, the new apache2.conf file that was added when I updated my web server had the following lines in it (my document root is /var/www):

<Directory /var/www/>                                                           
        Options Indexes FollowSymLinks                                          
        AllowOverride None                                                      
        Require all granted                                                     
</Directory>

I replaced the AllowOverride None line with AllowOverride all, restarted apache, and the permalinks now work.

Hopefully this will aid someone in the future with a similar problem.