It's one of the eastern districts and indeed the central part of the eastern side. And it has one of the biggest towns on the eastern side and the difference between western and eastern is you will see the settlements on the eastern sides are on the mountainside rather than valleys on the western side.
Mongar is about 194 km from Bumthang. The drive will take you through conifer forest with rhododendrons beneath and will cross over the highest point on Northern East-West Highway to Thrumsingla (3800 meters) and also will be at the lowest point on the later high way too at 650 meters at Kuri chu. So you will be crossing through a lot of vegetation and climatic zones.
And here you will not see people speaking national language, the eastern side have a popular dialects thats, popularly known as Sharchop-kha but its actually Tsangla-lo