What's the Difference between an Azure Cloud Resource and a Cloud Service?

Cloud Services are ways to combine a set of stateful virtual machines (or web/worker role instances, which are stateless virtual machines) into its own private network, sitting behind a public virtual IP address. The Cloud Service also provides optional load-balancing between the virtual machines). Cloud Services have been around since the beginning of Azure, and are considered part of the "classic" compute model. All classic virtual machines (whether stateful or stateless) live within a Cloud Service.

I don't know what you mean by Cloud Resources, but... Azure now has the Azure Resource Manager (ARM), which is a newer way to create virtual machines. There is no notion of a Cloud Service with ARM; the approach is to build out a set of resources (such as virtual machines, networks, NICs, IP addresses, etc) and dependencies, as a single managed set of resources.

The Azure site has documentation for classic Cloud Services here.

And documentation for ARM is here.

Azure service is simply a computing 'service' provided through the internet (cloud) than on-premises servers.

Docs:

Simply put, cloud computing is the delivery of computing services—servers, storage, databases, networking, software, analytics, intelligence and more—over the Internet (“the cloud”)...

Webopedia

A cloud service is any service made available to users on demand via the Internet from a cloud computing provider's servers as opposed to being provided from a company's own on-premises servers

Azure resource is an instance of the Cloud service(Or its components.) When you pay for a service and use it for something it becomes a 'resource' for you.

Docs

A manageable item that is available through Azure.

You can also see the difference on the Azure portal when you click 'Azure Services' (It lists what they can provide) and 'All Resources' (It lists what you already have)

Azure gives you services like Compute, Networking, Storage etc.

Within each service are resource types, such as Virtual Machine, Virtual Network, Storage Accounts etc

When you create a Virtual Machine, it is a Resource.