Clients are customers or who you're doing work for. Contacts are individuals you may contact when performing a service for a particular client.
A client can have multiple contacts associated with them, and a contact can be associated with one or many different clients.
Clients and contacts have a parent and child relationship.
Example 1
You may be working with ABC theaters and installing speakers at two of their locations. The client would be "ABC Theaters," and the contacts assigned to that account could be managers from each location you may need to contact while performing the work.
Example 2
You may have a couple of clients for whom you perform similar tasks and services and use the same builder. For example, you may have an account with Acme, Zoogle, and NCG.
Since you use the same builder, you can add the builder as the contact to all three clients and have this person's information populate in the opportunities and projects on Cloud.