They are nearly interchangeable, with only small nuanced differences.
"We have met before" vs "We met before"
A few reasons to possibly prefer "We have met before":
1. "we met before" is brief and abrupt. Adding more words makes it sound more friendly.
2. emphasizing the positive nature of the statement. Have we ever met?? Yes, we have met before. (instead of haven't)
Comparing "I had my dinner already" vs "I have eaten my dinner already"
If someone were angry, they'd probably choose simple words, the simple past, "I had my dinner already". If the conversation were polite and cordial, they might say "I have eaten my dinner already, but we could still sit together." The present perfect sounds more refined. However, these differences are subtle. You could even swap around those two sentences, the "supposedly" angry and polite versions, and it would still be alright.