If you were able to only go to one mountain area which one would you pick
Bieszczady or Zakopane?
Well, hard choice, those are two different kind of the mountains actually... It depends how much you like trekking...
As a trekking type of a man I can offer a few hints:
a) If you prefer smaller version of Alps like mountains go to Zakopane and Tatra mountains
b) If you prefer more hills like mountains choose Bieszczady.
Ad. a) Tatra mountains in the summer are terribly overcrowded but there is a way to avoid this, but maybe a little chalenging. The thing you should do is to not even stop in Zakopane or maybe just for one day. Instead of that go with your backpack straight into the mountains and Tatra National Park. There is a couple of nice refuges there. So if you start trekking from the refuge early enough in the morning, the feel of emptiness, wild and beauty will be your reward. Personally I see no point in other approach to the Tatra mountains in the summer, there is just to many people. As for refuges there is probably no way you can book anything right now, but at least in last year, except from refuge "Piec Stawow", in every refuge you will find shelter but sometimes on the floor.. It is a hazard, but when in last year I was doing backpacking through refuges line, we get some beds everytime...
Ad. b) I've not been to Bieszczady for ten years but I've heard it tends to be overcrowded in the summer to in those days... But if you go there it is not a bad idea to spend a couple of days around Solina lake.