From their website:

"FlaskBB is a simple and extensible forum software which should be seen as the foundation for building communities. It is written in Python using the web framework Flask."

it's just too basic and not popular

I have no direct experience with Flaskbb. But I develop almost exclusively in Python mostly using the Bottle.py micro-framework. The Flask framework is a more complete framework than Bottle.

Too basic?
Anything you can do in PHP you can do in Python and much more. So I am not sure what type of out of the box functionalities you need more than what is offered but I am certain that you can write code in python/flask relatively easily to provide it.

Not Popular?
I am not sure how popularity figures into the equation. Flask is not as widely used as PHP but there is ample support for Flask on stack-exchange. In my mind using very popular solutions can make you more of a target for spammers, as they commit more resources to hacking popular platforms.

The company doing this job suggests using Flaskbb instead of phpbb.

Did they give you a reason for making that change? I'd certainly want something concrete before breaking something that is already working (or can be made to work). Just asking. I don't use any of these products, but when someone says "use this" I always ask "Why?"

The current forum is hosted on a very old server and uses a database which has many problems that can't be fixed. So we need to rebuild the entire site including the forum from scratch.