What's the difference between посылать & отправлять?
My dictionary tells me that both these words mean send, but is there a difference in meaning or are they synonyms?

Question

Answer

Они синонимы, но "послать" немного шире:
1) Послать сотрудника в командировку (можно "отправить").
2) Послать письмо/деньги/посылку (можно "отправить").
3) Послать привет (нельзя "оправить").

Я думаю, бывают ещё случаи вроде 3, но чаще всего эти глаголы взаимозаменяемы.

Answer

Depending on what  and where "посылать, отправлять", they aren't  only translated as "send".

Off the top of my head, I will give you some more examples:
отправлять (посылать) письмо почтой- to post a letter
отправлять (посылать) войска- to dispatch troops
отправлять (посылать) на тот свет- to kill
отправлять в ссылку (сослать)- to banish,to exile
отправлять (посылать) в нокаут- to knock out

Answer

How would you say "I send him something", without being rude of course?

Answer

We must be very carefully with "send = послал" when we use it for person.  It's better to say "I dirrect him = Я направил его".   Because "я послал его..." is very same to "I say 'fuck you' to him"

Answer

They mean exactly the same thing. 
There are a few fixed expressions where you would use one over the other.
I can only think of the two that that are given in the answer above.
Отправить на тот свет, отправить в нокаут - here you wouldn't use "послать".

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