Phobjikha Valley

If you want to see the black-necked cranes, then you need to visit in winter when it is both very cold (and the accommodation basic) and reputedly inaccessible. If, however, you want to see the festival at Gangtey Goemba, then you are stuck with October (or September depending which year), so you can't have it both ways. That means unless you are rich enough to visit more than once, or a massive fan of Bhutan, you will only get to see one or the other.