One of them is adhikāra. Possible differences:

1. adhikāra is generally a vox media (you have the adhikāra to do X, which does not necessarily mean that X is a good thing), unlike “right” (where generally having the right to do X is a good thing).
2. adhikāra might imply duty, whereas rights don’t (you may have the right to remain silent, and this does not imply that you ought to remain silent). For instance, for Prabhākara if you have the adhikāra to perform a given sacrifice, you also have the responsibility to carry it out. However, it is not so in Kumārila or Maṇḍana, where the additional obligation to perform fixed rituals descends from their fixedness (nityatva), not from the adhikāra, as proven by the fact that no such obligation follows in the case of elective rituals (kāmya).
3. adhikāra is connected to ability (sāmarthya), whereas this does not apply to rights, which can instead ground the need of ability being ensured. For instance, if you have the right to go to school but cannot physically move, (in an ideal case) your government will provide you with the devices needed to let you attend school etc. By contrast, adhikāra presuppose ability, in the sense that unless there is ability to do X, there is no adhikāra to do it. Since adhikāra is a vox media, this might be a good thing after all. For instance, if you don’t have the adhikāra to do something difficult to get A, you will be allowed to do something easier instead. Please notice also that Śabara helpfully distinguishes the lack of an external (bahirbhūta) ability (sāmarthya), which is temporary and does not affect the adhikāra (for instance, you don’t lose your adhikāra if you temporarily run out of ghee or are too poor to perform a given sacrifice), and intrinsic (ātmavṛtti) ability (śakti), in the absence of which there is no adhikāra. Much to my disappointment, this distinction is not kept by later authors.