(1) Osho organized an order of his followers, which admitted both Christians and Hindus.
(2) Osho organized an order of his followers that admitted both Christians and Hindus.
These sentences are both correct. My intuition is that (1) is more likely to be applicable, but there are circumstances where (2) could work better. As you surmised, there is a difference of emphasis. "Which" usually[*] gives you more information about the thing that's been mentioned, while "that" introduces a restrictive relative clause that defines or restricts the scope of the item referred to.
([*]Some people use "which" for both types of relative clause, leaving the distinction between the two down to whether a comma is present, and whether the author has punctuated correctly. On the other hand, it is definitely non-standard to use "that" for both. "That" should always be used restrictively.)
The use of "that" in (2) might suggest that he organised more than one (and this one admitted both Christians and Hindus) or that he had previously organised one that didn't admit both. ("Osho had initially organised a Hindu-only order. But later he was to organise an order of his followers that admitted both Christians and Hindus.") Alternatively, it might just highlight the surprising nature of the new information. ("Osho was believed by some people to be anti-Hindu. But he organised an order of his followers that admitted both Christians and Hindus.")