One difference between standard images and cloud images is that Ubuntu Cloud Images come with cloud-init
From the first link : "Ubuntu Cloud Images are pre-installed disk images that have been customized by Ubuntu engineering to run on cloud-platforms such as Amazon EC2, Openstack and LXC." So, that's what those are for.
The second link is for their normal distro. If you wanted to install Ubuntu on a physical (or virtual) machine that you are sitting in front of or otherwise have access to and good control of, you would download something from the second link and install it.
Ubuntu cloud images dispense installing to a (virtual) disk from the ISO
When you install a desktop ISO, it takes you through an interactive installation setup that sets things like partition sizes, username and language settings. You can see such a setup at: https://askubuntu.com/questions/884534/how-to-run-ubuntu-16-04-desktop-on-qemu/1046792#1046792
This is however too inconvenient for Cloud deployments, which require spinning up a large number of OSes, and things have to be automated. This is why the Cloud images exist.
In particular, as of 18.04 it ships a pre-installed qcow2 image that you can just boot out of the box without the installer. This image format can also be easily resized.
These images are also very useful for emulation if you just want to get Ubuntu up and running quickly, I have shown a QEMU setup at: https://askubuntu.com/questions/281763/is-there-any-prebuilt-qemu-ubuntu-image32bit-online/1081171#1081171
Not sure that the first answer is correct.
cloud-init is the Ubuntu package that handles early initialization of a cloud instance. It is installed in the official Ubuntu live server images since the release of 18.04, Ubuntu Cloud Images and also in the official Ubuntu images available on EC2.