the analogy is usually sunni:shia::protestant:catholic. in A History of Iran: Empire of the Mind i stumbled upon another one, sunni:shia::judaism:christianity. the argument seems to be roughly that judaism is a legalistic religious focused on orthopraxy (at least in its historically dominant rabbinical form), while sunn islam is defined by its four madhhabs which define the bounds of sharia. in contrast, christianity is a more spiritual demonstrative religion focused on particular aspects of belief which exhibit emotional salience. in particular, the central role that karbala plays and emotional mass rituals triggered are analogized to the passions which have been common throughout european christianity (though interestingly these displays are attenuated in protestantism). i also think that one might note that to a large extent christianity and shia islam arose as sects at odds with the state, and so to some extent are habituated to be suspicious of secular power and naturally be more open to the idea of disintegrating and separating the state from the religion more naturally than sunni islam, which to a large extent was defined by lack of dissent from state islam.