Take note of peak & non-peak periods
The peak tourist seasons are Spring: March - May and Fall: September - November. In particular, March, April, October and November are especially busy. If you're considering visiting during these months, it's best to book your tour package and flight tickets at least 3 months ahead to ensure your reservation.
If you want to avoid the crowd of the peak tourist period, consider travelling during the non-peak period.
Know the seasons
It really depends on what you want to see and do. If you want to trek, April, May, September and October are the best months with optimum weather. Although it is colder, the skies are generally clear and blue, and most importantly, it won't be muddy. Do note that the monsoon season in Bhutan is from July - August and during this time, light rain (1-2 hours) in the morning is typical. Heavy downpours are rare, but we generally do not recommend trekking, unless they are short day hikes.
Winter is a good time to catch the endangered black necked crane in their winter home, the Phobjika valley and summer is a wonderful time for mushroom picking (there's even a Matsutake Mushroom Festival) and to catch glimpses of (sometimes double) rainbows over the valleys. If you're there to seek solace or for spirituality reasons, anytime would be a good time to go.