In the beginning, let's note that Cloud and serverless solutions don't mean the lack of a machine because somewhere at the end of the rainbow, there is always a physical server. However, the most significant difference between AWS or Azure advanced hosting services, and a dedicated server is the approach to development.
As a rule, you are relieved of many administrative issues with Cloud-based solutions. For example, if you don't want to, you don't have to be interested in how many physical machines you have, what kind, and how to maintain them. Likewise, you don't have to worry about installing the database service, scaling the server manually, or doing backups.
With Cloud-based solutions, you get access to an admin panel and console utilities with multiple ready-to-use services that can make your work easier. Lambda, a tool for building serverless microservices, is a great example. Even if you know little about AWS, with Lambda, you can run code for any app with minimal configuration overhead.
Suppose you wanted to do it on an on-premise server. In that case, you'd have to configure all the required services and infrastructure, where some operations like scaling might be significantly challenging to implement correctly.
Lamda is just one example, which presents the benefits of using Cloud-based services since it's one of the closest to the 'serverless' approach, but there are many more. In AWS (the Cloud platform we use most often in OG), you can choose from several services to run a database (relational or NoSQL), a service for user authorization, multiple utilities for video streaming, or a CDN for serving files and caching your web applications.
You don't have to worry about many standard administration routines related to doing these things on your own. And since you have all these services within one architecture, you get minimum latency times, too (at least, that's what we want to believe in).
