Прошу is an answer for "thanks" (like "you're are welcome"), you can't use it to ask for something. Будь ласка can be used both ways
My family speaks pre-WWII West Ukrainian, we use прошу to mean please. However, from what I've heard, it seems like only Будь ласка is used in modern Ukrainian.
In contrast, we use "proszę" in Polish for both "please" and "you're welcome", but there is also rather archaic "bądź łaskaw" which can be used only for "please" :)
not the word to word translation, but it is close in meaning to "Would you be so kind to". Today "будь ласка" is a shortened form from old "Коли буде ваша ласка" or "З вашої ласки". "Ласка" is indeed "endearment"
PLEASE, I'm not afraid to ask a question... If I get BOOED at i'LL ASK Again
Explanation by JewishPolyglot:
I don't know why people down-voted my comment below - it was just a way for me to remember that будь ласка means please. Look at the capital letters and reconsider your vote