While both can be grammatically valid in a sentence, I think when using "I" to refer to yourself, you refer to your whole character, your self-image, your identity. Using "me" is closer to "this person here"; it is less ... hm, self-aware, if you will. It distinguishes you from others, but not much more.
As an example, I would like to quote a scene from - yes - Star Trek, where a high-ranking, adorned and much respected embassador makes the official introduction of his wife with the words
May I present she who is my wife, (name).
Of course, this is a little more elaborate than just replacing "me" with "I", but it goes into the same direction.
PerryW's "It is I" and "It is me" is a very good example. They both say the same thing, identifying the speaker in response to a question. If that question was, say,
"Who threw that tomato at the speaker?",
they are both valid to identify you as the thrower. However, you would only proclaim "It was I who threw the tomato!" if you are already halfway onto the stage, with a speech of your own explaining your own honest and heartfelt reasons for this offensive act. There is a lot of pathos in saying "It is/was I".
If, however, like most people and tomato-throwers, you just want to make the speaker shut up and leave, a simple "That was me alright!" will do. Or if you immediately regret having thrown it but can't steal away, you may just mumble "me", but not "I".
The situation is the same in the beginning, the message is the same, but using "I" instead of "me" will change how people perceive your answer. In any case, the nominative "I" (or "we"/"she") is very rarely used.