Upsides of lambdas are ease of deployment (no need to worry about servers, that's kind of the whole point of serverless), virtually infinite horizontal scaling, and a very generous free tier.
Downsides are relatively slow cold starts, difficulty of exposing to the outside world (eg via HTTP route), and lack of state management.
Personally I like using Lambda as glue between different parts of the AWS ecosystem, or to handle events, dispatch notifications etc.
However I would definitely not use Lambda for anything remotely resembling a stateful web app, for instance. The slow cold starts and inherent statelessness are going to make that difficult. Also API Gateway is a huge pain to work with, or was last time I looked at it.