I have used MyBB,phpBB,vBulletin and Xenforo.
I preferred vBulletin overMyBB and phpBB for a long time.vBulletin has an excellent ACP and very customizable with a ton of mods.
However Xenforo is the best forum platform I have ever used.You can look at many sites and spot Xenforo immediately.It is very customizable with an excellent ACP.It also comes with a lot of useful features and a good amount of mods.
Of course MyBB and phpBB are free open source forum platforms and sometimes these are more desirable to use when starting a new forum.Once your established and want to move to a paid forum platform I would suggest Xenforo.
Between MyBB and phpBB depends on which you prefer.phpBB has been around longer than MyBB however MyBB had a major upgrade about 5 or 6 years ago with a lot of new features.Personally my choice would be MyBB,I believe MT was on MyBB before Xenforo but I could be wrong
I would stick with what you like and what you are comfortable using because the that should be the least of your worries starting a new site.
I wish you good luck with your project and if you have any questions feel free to ask