"Je vous en prie" appears to have two meanings:
(1) literally, "I beg/pray (the aforementioned thing) of you,"
I beg you to help me..... Je vous en prie aidez-moi.
God, help me, I pray / I'm begging you ..... Mon Dieu, aidez-moi, je vous en prie...
(2) "You are welcome"
"de rien" would be more likely to be used when chatting online or speaking to someone close like family or friends, almost in passing, or as a spontaneous response to spontaneous thanks:
"Here is your drink.
"thanks"
"you\'re welcome" = "de rien"
(although "je t'en prie" would not be wrong, just less relaxed).
"Je t'en prie" is similar - it uses the "tu" form for "you", rather than the more formal "vous" - but would be more likely used by someone slightly less familiar or more formal in their style of speech, or by an adult to a child, or when thanking someone more specifically, rather than in passing.
"Thank you so much for all your help today"
"You're welcome" = "Je t'en prie"
"Je vous en prie" is more formal (using "vous" rather than "tu") and would most likely be used when dealing with someone you don't know well, or at all - in a shop or when dealing with someone by phone etc...