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you're on a date

A: Tell me something interesting about yourself.

B: I've climbed Mt. Everest/ I climbed Mt. Everest

Are both correct? What's the difference? Is it a matter of choice or is one of them wrong? Thank you.
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It is a matter of choice, though the difference is not great. "I climbed Mt Everest" is a plain statement about a past event. "I've climbed Mt Everest" reinforces the idea that this past achievement contributes to your present state, in terms of the experience that you now have.

Comments  
NecrophagistAre both correct?

Yes.

NecrophagistWhat's the difference?

Not much. Maybe the simple past is a bit more forceful, but it's a toss-up.

Necrophagistis one of them wrong?

Strictly speaking, the present pefect is designed to convey the required meaning, but people rarely observe the distinction in my US dialect and use the simple past. There is a joke about our speech:

Bob: "Jeet jet?" (Did you eat yet?) (In favor of the more proper "Have you eaten yet?")

Dave: "Nah. Jew?" (No, did you?)

 GPY's reply was promoted to an answer.