两者意思有重合处 都有“逐渐消失 平息” 的意思 主要是看所使用的语境了
die away：强调的是（情感,风、声音）减弱/平息；还有动植物的死亡,消失,以及 假死（昏厥）这个用的极少
His anger died away.
他的怒气逐渐平息下来了.
The winds die away after the typhoon passed.
台风过后,风渐渐平息了.
The noise of the car die away in the distance.
汽车的声音消失在远方
Almost as quickly as they appeared,these small plants die away.
这些小植物很快就消失了,几乎就象它们长出来时那样迅速.
She died away at the sight.
她一看到这情况就昏过去了（这个少见）
fade away 强调的是 图像,情感方面的东西“ 逐渐消失褪色”比如说记忆 ,人的感情,荣耀等
Memories of my sister faded away after many years.
过了多年以后,对我姐姐的记忆渐渐淡化了.
The memory of his hard childhood will never fade away.
他童年时期的痛苦记忆永远不会逝去.
The wins and losses that once seemed so important will fade away.
而曾经对你如此重要的成功与失败也将随风飘逝.
Even the glory had fade away,but you can still shine if you meet the right sun.
荣耀有退色,但你能仍然发光如果你遇见正确的太阳!
当然两者也有重合的地方 比如表示声音时候两者都可以使用.但是其它的方面还是有很大区别的 希望对你有所帮助