I'm not that experienced in forum scripts, therefore I'm probably not the right person to give some recommendations.
Some years ago my first forum was also a phpBB. Since I discovered some annoying behaviour in that forum, it's no longer my preferred script. I normally don't register with a phpBB forum. Lunascape's forum is one of a few exceptions.
Aren't there any plugins or patches for phpBB to make it resistable against automated bot account creations?
What about
XMB eXtreme Message Board? I had one installed for testing purposes. But it's also a few years ago. It's quite good, it runs on php and it's free. But I don't have any idea about resistance to spam.
SMF is well known, runs on php and is also free. But I've never installed and tested it.
vBulletin or IPB are both not free, but the first one offers quite a lot of features.
If the script does not have to support MySQL then
MyUPB could be an alternative. It uses flat files for the database.