The tl;dr is that Bhutan was not legally a princely state, but was effectively treated as one. If I can quote Karma Phuntsho's The History of Bhutan in full:

King Ugyen Wangchuk, it appears, was fully in favour of the British proposal. On 8 January 1910, the treaty of Punakha was signed and Bhutan agreed 'to be guided by the advice of the British Government in regard to its external relations'. We don't know how far Ugyen Wangchuk and his council understood the political implications of this article in the treaty and also other panegyric letters of submission they wrote or statements they made to the British. But Charles Bell took this much further than the literal sense of words and declared: 'Thus, with the consent of all and with the blessing of its all-powerful priesthood, Bhutan was joined to the British Empire.' Generations of Bhutanese would deny that Bhutan ever was under the British empire and the British, on their part, also remained equally muddled about Bhutan's status. In their official administration, the British neither considered Bhutan to be under their rule nor gave Bhutan the kind of benefits received by princely states in the region. Today, it is not part of the Commonwealth of Nations. However, their generation treatment of Bhutan was not very different from Indian princely states. In fact, even before Ugyen Wangchuk became King when he travelled to Calcutta in 1906, the British showed him the same reverence which they showed to a mid-level Indian prince with a salute of fifteen guns. He seems to have happily accepted this status and vaguely viewed Bhutan as one among the array of nations under the supreme British government.

The Kings of Bhutan were usually referred to as Rajas when in the context of India and were treated similarly. The main difference stems not just from this 1910 agreement, but from the Duar War of the 1860s when British India signed a treaty with Bhutan granting the Bhutanese government a stipend in exchange for not raiding into Indian territory.