Dedicated Servers:
A physical machine in a specific location is dedicated to running server code. 100% of the physical machine's resources are dedicated to running the server code.
Cloud:
Huge data-centers spread all over the world run server code on VIRTUAL machines and worker roles. Virtual instances of the server code can be spun up or spun down at any data-center around the world within minutes. Scaling is automatic, and dependent on how many users hit the service.
Dedicated servers really have no benefit over the cloud, but the cloud has tremendous benefits over physical/traditional dedicated servers.
Most games which have traditionally had dedicated servers have no where near the vast scale that Azure's cloud has.
The Cloud is more cost effective.
Dedicated servers are expensive. Prohibitively so for independent game makers.
The cloud compute SDK for Xbox One devs is free.
The Cloud never 'dies'.
Dedicated servers eventually need to be shut down, and when they are, the service they were providing goes away too.
With the cloud, since virtual machines can spin up or spin down as needed, you could (in theory) have a service that never dies; it would just scale down dramatically as use scaled down... but as long as the "cloud" that hosted it was around, the service would stick around.
The Cloud provides lower pings for more users world-wide.
Devs which have relied on dedicated servers in the past has usually only been able to afford a few server farms to host their dedicated servers. The further away from the dedicated servers you are, the worse your lag/latency/performance. With the cloud, Azure is expanding all the time, constantly. It has well over 300,000 machines spread over data-centers all over the world... no matter where you are in the world, if you're connecting to the cloud, you'll probably have pretty good pings.
The Cloud is more robust, resilient, and nimble.
Dedicated servers that crash out take more time and energy to restart, and when one machine dies - generally overall capacity is hit until that machine is restored. With the cloud, if a VM (virtual machine) dies, another can be spun up in minutes, and you're back up and running.
The cloud is just vastly superior to dedicated servers.