The expression is "for the skeptic, no proof is sufficient; for the believer, no proof is necessary" exemplifies one such collision of worldviews.

A more specific example would be an exchange between somebody who believes they have been healed by a herbal remedy, trying to convince a skeptic. The skeptic believes that a person can be persuaded to interpret symptoms generously in favour of what they expect, whereas the cured person considers it an insult to be accused of bias, believing that what they observed is incontestible.

A more common collision of worldviews exists between cultures. The belief in affirmative action seems to have a lot less appeal to white males, for example.

I'll flesh this section out more over time, but right now it's hard getting the material organized because it's a fairly new field of study