The best replacement for phpbb is vbulletin.

The best alternatives to phpbb are rated as following:
★★★★★  vbulletin
★★★  smf
★★★  vanilla
★★  mybb
★★  ipb
★★  bbpress
★  invision
★  simple machines
★  kunena
★  fluxbb
★  xenforo
  buddypress
  yabb
  ip board
  punbb
  phpnuke
  simple machines forum
  discuz
  fireboard
  php nuke
  simplemachines
  smf forum
  proboards
  phpbb2
  discourse
  yaf
  phorum
  invision power board

Your alternative for phpbb is not listed? Suggest!
Speaking about phpbb, one probably shouldn't forget about joomla.

Other search terms related to phpbb and their ratings are:
★★★★★ joomla
★★ wordpress
mambo
You have in mind a companion for phpbb and it is not yet here? Propose!



Google Trends

Scale is based on the average worldwide traffic of phpbb in all years. Click on graph for more.
phpbb
vbulletin
smf