The business case I give is a website which has a predictable spike in traffic which tails off.
In the UK we have a huge charity fundraising event called Red Nose Day and the public can donate online (or telephone if they want to speak to a volunteer).
The website probably sees 90% of their traffic on the day itself - millions of users - and the remaining 10% tailing off a few days later. Then nothing.
The elasticity of the cloud allows the charity to massively scale their compute power for ONE day, then reduce it for a few days, and drop back down to a skeleton infrastructure until the next event - in a few years time.
(FWIW I have no clue if Red Nose Day ever uses the cloud but it's a great example of a business case requiring temporary high capacity compute to minimise costs)