Bhutan has some of the best trekking in the world, with breathtaking scenery, a rich Buddhist culture, friendly people, and few other visitors.
The difficulty with trekking here, aside from the high cost ($200/day), is the weather. Unlike Nepal and Ladakh, the windows for good high-altitude trekking in Bhutan are short: about 3 weeks in April and 3-4 weeks in October. Although the official season extends longer, you have the best chances of clear weather and little snow in the second half of April or the first three weeks of October. There are several low-altitude "winter treks" however, which can be walked in other seasons.
Although most of the treks themselves are not difficult, the high altitude and difficulty of reaching rescue services means that especially the treks through the High Himalayas are not for the faint-of-heart. However, if you have saved up some money, and are in good shape, you will have one of the best experiences imaginable.