A political party in Bahrain is called a society. What are the fundemental differences with political parties in Western Liberal Democracies, in the sense that the word society is used?
In essence, the word phrase "political society" seems to be used for the simple reason that political parties as such are illegal in Bahrain. The "Political Societies Act" of 2005 seems to have legalized political societies, but not parties. Although I can find references to this, they're all from 2010 or later.
As for the differences between these societies and traditional political parties, they have less to do with the structure of the societies themselves, which seem to basically be parties insofar as they have members, sets of specific political beliefs, and leadership structures, and their members can attain political offices. See for instance the example of Al-Wefaq, which has an organizational structure, a platform (Islamist) and elected representatives. Rather, the differences have more to do with the extent to which they can influence the government.
Most obviously, in a democracy political parties have a chance to control the executive power. Regular elections take place, which are meant to ensure turnover between parties and executives in accordance with the will of the people. The prime minister, president, or other executive is selected by the population in some fashion, either through direct election or through some form of representative, such as an elector or a member of parliament.
In Bahrain, this is not the case. The Prime Minister has been the same person for 48 years—the entirety of the existence of the nation in its modern form. That current Prime Minister, Khalifa bin Salman Al Khalifa, also happens to be a member of the royal family. The prime minister is appointed by the monarch, who has broad powers. The current ruler is Prince Salman, who has been in charge for twenty years. The sole governmental organ to which people can be elected, the Council of Representatives, has little power to contradict the king, and parties are subject to dissolution if they oppose the government.
As a side note, the phrase Western Liberal Democracy is a bit misleading, insofar as it often implies that non-Western hemisphere or non-European democracies aren't necessarily actually democratic or liberal. There are various liberal democracies that are not in "the West", such as India and Japan, and many more democracies generally. So this answer deals with democracies or liberal democracies generally.