I heard the following from a radio show, in which the hostess said (something about the recent flood in Beijing ), "我们的责任和希望不在于尽力而为，而在于全力以赴".
In my opinion, both idioms mean to go all out or try everything one can, but why here do they become so different that the speaker uses "not..., but..." ?
尽力而为：try your best
全力以赴：whatever it costs, do it
力 in 尽力而为 refers to 能力 (ability); 尽力 (exhaust one's ability)
力 in 全力以赴 refers to 力量 (power); 全力 (with full power)
尽力而为 = Do the best within one's ability
全力以赴 = Do it with the utmost earnestness; go all out
尽力而为 is more reserved compared to 全力以赴 which is more energetic
Examples:
对方是世界冠军，这场比赛你们尽力而为就够了 - The opponent is the world champion, just do your best within your ability is enough (not to care about win or lose)
对方虽然世界冠军但我们也要全力以赴争取胜利 - Although the opponent is the world champion, we must go all out to win (try hardest to win)
More examples:
這次任務成功率只有一成，但我会尽力而为 - The success rate of this mission is only 10%, but I will try my best (expect to fail)
這次任務成功率只有一成，但我会全力以赴 - The success rate of this mission is only 10%, but I will go all out to do it (hope to success)
The difference between these two is about the attitude toward something with low confidence:
尽力而为：Because I am not confident about something, all that matters is that I have tried my best. Failing is not shameful.
全力以赴：Even though I am not confident about something, I will try to achieve it at whatever cost. Failing is not an option.
Imagine you are a soldier fighting a war, at a disadvantage. 尽力而为 would mean trying your best to fight back enemies but you can retreat if you know you can hardly win. 全力以赴 would be accomplishing your mission at whatever cost even if it means dying in the war.
So in some contexts, 尽力而为 would imply "I may fail so don't blame me if I do", 全力以赴 on the other hand implies you are determined to accomplish the task even if you know it's difficult.
全力以赴 means put all your strength into it, that is, the last stand and there is no way out.
尽力而为 means to do one's best and to do it by one's own strength.
They all mean taking a positive approach to something that is very difficult.
But their emphasis is different.
全力以赴 is a kind of desperate posture, to achieve the goal at all costs. Focus all your energy on one thing.
尽力而为 is to take it with a grain of salt, to do your best. It means you're doing the best you can, and it doesn't have to cost you much.
To me, they are exactly the same. No difference.
You can look the terms up in a thesaurus and probably find them listed as synonymous.
We actually use and interpret the two idioms in different ways, if not opposite. But no matter what, they are not exactly the same.
(1) The attitude of the speaker.
尽力而为：I will do my best to make it happen. (I am an outsider, but I am willing to help.)
全力以赴：I will put whatever power and resources that I have into this objective. (I have the obligation to do it.)
(2) How honest the speaker may be.
尽力而为：I will do my best. (I am honest but it may not happen.)
全力以赴：I will do it with full strength. (most likely, a political slogan.)
For example: (In the ICU of a hospital, the doctor may say:)
我们会尽力而为。（to an average Joe）
我们会全力以赴。（to a high level VIP）
尽力而为：do one's best 全力以赴：spare no effort to do
