Present participles (e.g., meeting) express that an action is taking place - that it's been actualized, whereas infinitives (e.g., to meet) express that an action hasn't happened yet - it's unactualized. That's why "to meet" is used as a greeting and "meeting" as a farewell,
 Nice to meet you. <introduction>
 Nice meeting you. <closing>
Hope that helps.
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