Sorry this is late, but I might as well offer my 2Cents.

Geography, as has been mentioned earlier, is a big factor here. Bhutan is on the southern slope of the Himalayas. Indeed, it's actually shorter from Lhasa to Calcutta than to major Chinese cities, so even Tibet proper had a historically bigger trade network with India than with China (this can get a bit sticky, "Tibet" usually refers to in historical context, the Tsang area, the Lhasa-Shigatse axis, not Kham or Amdo which are more nomadic and tribal regions that would have closer and easier trade with western areas of China).

Anyway...

Historically, Bhutanese trade has a more north-south dimension with Tibet owing to cultural and historic ties. Bhutan formally split from Tibetan control (though informally, there wasn't a lot of control exerted throughout the centuries) in 1616 when the Zhabdrung Ngawang Namgyal migrated there and began building a number of castles across the eastern Himalayas essentially amounting to the country you see today. The one exception to this was a region of foothills along the southern border known as the "Duars" or sometimes historically the "Doors."

A lot of this area was subject to Bhutanese tribute and raid, mostly owing to whoever the strongman was at the time, culminating with the "Black Regent," Jigme Namgyal. The Bhutanese wanted this area because it was a huge source of their economic power. The British eventually put their eye on the region because the Himalayan foothills is an excellent area for growing tea.

In 1863, Ashley Eden was appointed as the British Envoy to Bhutan. Eden led the invasion of Sikkim in 1861 which resulted in a favorable treaty with the Chogyal of Sikkim. Eden took his mission to Punakha and was supposed to negotiate a subsidy of 2000 Rupees (or a sum equal to one-third of the total revenue annually) assuming that Bhutan did not raid into the Duar area (which the British assumed they owned as a part of their treaty with Assam, which no authority in Bhutan recognized). Areas of the Raj as well, namely Cooch Behar and Sikkim, that Bhutan often regarded as its own sphere of influence (i.e. subject to raiding, religious rights, etc.) were forfeit.

Jigme Namgyal (the father of the first Druk Gyalpo of Bhutan, Ugyen Wangchuck) sent Eden on a wild goose chase and a set of cultural humiliations (one particularly hilarious episode involves rubbing dough on Eden's face). Eden left Bhutan with the conclusion that the Bhutanese were barbarians, and that Jigme Namgyal was, "an avaricious, treacherous, and unscrupulous robber." The British went on to invade Bhutan, conquering Punakha, but then had some trouble getting out. The Treaty of Sinchula was signed on 11 November 1865 which agreed to suspend hostilities providing 1. perpetual friendship between the two countries 2. permanent British annexation of the Duars 3. release of British subjects in Bhutan, 4. a subsidy of 25k Rupees, rising to 50k in three years, 5. the right of the British to withhold payment if Bhutan failed to check future frontier outrages, 6-7. mutual restitution of criminals, 8. arbitration of the Government of India in disputes between Bhutan and the British protectorates of Cooch Behar and Sikkim, and 9. free trade.

The treaty was continued until 1910, when the British and the new Kingdom of Bhutan signed the Treaty of Punakha (essentially an updated extension), and then 1947 when the new Republic of India signed a new treaty of friendship with Bhutan, which was mostly a continuation and extension of the Punakha Treaty. Between the Duar War and Independence, Bhutan essentially had observer status in the British Raj and was treated as one of the Maharajas. When the Kings of Bhutan visited India, they were seated at the same height (extremely important) as the Maharajas and were introduced along with them. Although trade was still mostly north-south with Tibet.

In 1950, that all changed when the border suddenly closed with the Chinese invasion of Tibet. The trade did a complete 180, and trade became a big commodity with India. Today, India is Bhutan's biggest importer, primarily of hydroelectricity (notice the rivers and that aforementioned southern slope).

The above information on the Duar Wars was an extremely abridged piece of the Duar Wars chapter from The History of Bhutan by Karma Phuntsho. Though there are several works, mostly British, on the subject.

We actually had a resident Nepal/Himalayas expert around at one point that did an AMA about a year or two ago, but I can't seem to find in the archives who it was.

However, I think a big piece of this is simple physical geography. If you look at where Bhutan is in the larger geography of Himalayas, you see that it's on the Southern flank of the mountains. To the north is the tibetan plateau, which historically was not culturally part of China Proper. To the South is the foothills of the Himalayas and the floodplain of the Ganges and Brahmaputra rivers. It's much easier to communicate with and travel to the major population centers of India from Bhutan than it is to travel to the major population centers of China.