Difference between these two phrases for "Killing two birds with one stone": 一箭双雕 vs. 一石二鸟
As you mentioned, both 「一箭雙雕 」and 「一石二鳥」are equivalent  to the English idiom " kill two birds with one stone" . The difference is the scenario of the metaphor . You can say the two idioms are basically the same, but「一箭雙雕 」 is more decisive and forceful, while 「一石二鳥」 is more methodical and calculated.


「一箭雙雕 」(one arrow goes through two condors) describe two goals being achieved by one straight forward action-- that take great aim, skill and power.
「一石二鳥」( hit two birds with one stone) also describe one action achieves two goals, but the stone has to hit the first bird and then bounds to hit the second bird-- that take skill, planing and calculation. 
「一箭雙雕」focus on the result: You can describe someone dating two sisters at the same time as  「一箭雙雕 」, but not 「一石二鳥」.
「一石二鳥」 focus on the method: People say " 「一石二鳥之計」( the trick that kill two birds with one stone) but not 「一箭雙雕之計」。
Another term similar to the two is 「一舉兩得」( take one action and get two benefits)