What is the difference between e.g. and i.e.?

Both are Latin abbreviations. E.g. is exempli gratia, for the sake of example, and introduces one or more instances from a larger set that you are not fully listing. I.e. is id est, that is, and introduces a restatement or precise clarification of exactly what you mean. The distinction is between illustrating and defining. Consider the difference between citrus fruit, e.g. lemons and limes, which implies others exist, and my favorite citrus, i.e. the Meyer lemon, which specifies the one thing meant. Substituting one for the other changes the meaning rather than merely the style.

The memory aids that survive contact with actual writing: e.g. is example given, i.e. is in essence. Both take periods after each letter in American usage, and both are conventionally followed by a comma. Neither is followed by etc., since e.g. already implies the list is incomplete and adding etc. is redundant. In formal writing, many style guides recommend confining both to parenthetical material and using the plain English phrases in running text, which sidesteps the confusion entirely and reads better besides. Related abbreviations that get similar treatment: etc. is et cetera, and so forth, and should never be written ect.; et al. is et alii, and others, used for people rather than things, with the period only on al.