The difference is in REGISTER. I think many native English speakers forget that we have register, like most any language. The transition "for instance" is considered to be in a higher register (more formal) than "for example" (more casual). So, while the meaning is the same, the MOOD is different. You might compare this with the difference in mood between the following: "You are ignorant and rude!" (lower register) "You are ignorant; moreover, you are rude." (same meaning, higher register)

The two idioms mean essentially the same thing. You might see "for instance" hyphenated and used as a noun: "Let me give you a for-instance." = "Let me give you an example."