Yeah, that monitor does not appear to offer a DisplayPort connection, which Nvidia currently requires to support adaptive sync on a FreeSync display, so if you went with the Nvidia card, you would lose FreeSync support.
I actually missed the part where you mentioned having bought the card already. Performance will vary by game, but on average, a 2060 would only be slightly faster than Vega 56. That difference probably isn't going to be noticeable at 1080p, where you should be getting good frame rates already. The 2060 does include hardware to accelerate raytraced lighting effects, but enabling those tends to cause a big hit to performance in the few games that currently support them.
As for how hot it will run or how audible the fans might be, it's difficult to tell without reviews. A 2060 does draw significantly less power than Vega 56 under load, so it should put out less heat, but the cooling system might potentially be significantly smaller as well, depending on the model.