Don't know about the invisionfree stuff, but I think it's not allowing you FTP access (or does it?). You can go with phpBB and not have an issue (I am running 4 phpBB forums and I love them all), but you need to have full access to FTP and the phpmyadmin section. So that you can install a lot of nice mods and customize your skin beyond recognition
vBulletin is a paid forum script but it is the best one that there is out there (in my opinion at least).
If you can shed a couple of hundreds dollars, it is worthwhile getting.
If you're poor, or if you are a free software zealot (I am -- I want the ability to freely modify & use without restriction), then you should use phpBB. It's low-end, but you can apply mods to make it high-end.
If you're poor and NOT a free software zealot, you could try SMF. The license is too restrictive for my tastes, but it costs nothing, and it's better than phpBB out of the box.
For everything else, use VBulletin. In particular, if you can hold out for 3.6 and start with that, that would be best. 3.6 onwards looks nearly irresistible. But it costs $160, as Dani mentioned.
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