Lambda is a focused, specialized service for running short-lived processes triggered by events and is one of the many services provided by AWS that have formed the foundation for many companies and other PaaS providers for many years, while Azure Mobile App Service is a packaged and branded collection of services including data storage and push notifications that is similar to backend service providers like Parse and Urban Airship pre-pivot (at least one of which was built on top of AWS), and tries to offer services comparable to a subset of other AWS services including SNS (push notifications), RDS or DynamoDB (databases as a service), among others.

It's also quite strange to spotlight Azure as if it's doing something at all remarkable considering the "Mobile Backend as a Service" market has existed for years and actually seems to be on the decline, at least as a standalone segment.