The lessons say they both mean "thank you." When should each one be used?
It doesn't matter when, you can use both. But the Russian mostly say "спасибо". So you wouldn't make a mistake.
I'll just add that Я благодарю тебя/вас/Вас - I thank you. Благодарю as a short variant.
Yes, that's another possibility too.
Use "благодарю́" if you want to sound like a XIX century gentleman.
Do you think it's old-fashioned :D
I'd say yes. How often do people actually use "благодарю" and "прошу прощения" outside formal writings today?
I know a few people who use "благодарю" casually. I don't think it's that old-fashioned, it's just a bit more flowery but that's all.
I tend to use "прошу прощения" with those who are dear to me. Извините/извини is enough for strangers.
"Благодарю" is a kind of GOOD old fashioned :-)
In a live recording of the metal band Аркона I'm pretty sure I heard the frontwoman say благодарю.
благодарю is a bit more formal. I can say this way, when I have a conversetion with infamiliar man. But if you'll say спасибо, it will be right too.
when my Russian friends and I are being sarcastic to each other we'll say блаааагодарю вас. It's like saying oh geez thanks your royal highness. Kinda like mocking each other with the formality.
So for me its an outdated term I wouldn't even use with бабушка.
благодарю = verb , I thank you (for your kindness)
спасибо = noun, thanks (well actually it's an archaic shortening "god save you")
so благодарю as a verb is more involved / formal (i.e. you stress that it is YOU who is doing the thanking)
Thanks. Knowing or is a verb really helps.