Not everything big data is limited to the latest twitter or facebook.
Just to give a one example of many possible, think of any processing that involves millions of records daily, and you need to process them, aggregate, analyze, dice and slice across various attributes on daily/monthly/quarterly/yearly frequency. This could be a financial firm, a retail chain, or anything where a lot of transactions happen daily. And you might be surprised that it's not just large corporations or governments.
Also, when there are permutations and combinations, input size increases dramatically.
But the biggest benefit it offers is to be able to do all this at very low cost - and that's the key factor of it's rise and popularity. Sure, big corporations were able to do a lot of such heavy task because they could buy hardwares worth millions of $$. But big data (hadoop) suddenly makes small players equipped with the same tools as big players and that is game-changing.
I am not defending big data just because it's cool, but because for a small company like ours, we can now think of doing things that were beyond our reach because of heavy cost of adding more storage and more processing power. It's very easy to understand the importance If you are into data warehouse / data analytics industry. On the other hand, if you are just creating few web apps, you may find it hard to understand the big deal.