The most impressive thing (for us) that you could do with AWS Lambda is easily scale your functions without being worried about scaling.
For example, we [1] have an API that runs Lambda functions to extract the structured content out of the news article. At some point, we have very few new articles being discovered by our system (night time in the US, for example). Other time, there are ~100 new articles every second. And, we do not have to worry about our infrastructure: 100 Lambdas will be executed concurrently, and we'll pay for just 100 Lambdas like if they were one after one.
Also, for many startups/side-projects when you don't know what would be your workload in advance, Lambda can save you money. And, if you somehow have a huge workload your solution won't be crashing.