短暂是口语，转瞬即逝是书面语，成语。

转瞬即逝 和 稍纵即逝 both of them have an other feeling,when you wanna to express the feeling of regret and you wanna talk to some people to cherish something like:爱情转瞬即逝，请珍惜你身边爱你的人。 知道失去我才发现，机会稍众即逝。 ：P

短暂 is an adjective, you can say 有的东西很短暂, which means they don't last for long. 转瞬即逝 is a verb, which means " appear for a very short time and disappear for good". e.g 流星转瞬即逝。

短暂和稍纵即逝的意思都是形容时间很短，但是相对来说，稍纵即逝的时间要更加短暂些：Brief and fleeting time, both of described time is short, but relatively speaking, the fleeting time is short more and more.For example，（短暂）youth is short，time is short，i need to go；（稍纵即逝）rainbow is always fleeting. on the whole，you just remember that the time of fleeting is less than short is ok.

短暂是有相对性的，相对正常而言。转瞬即逝就是指时间极短。例如：他在30岁那年发生了交通事故，走完了短暂的一生。