The Client is a Company.  The Contact is a person who works for the company.  

NOTE:  For those that work domestic type cases or cases for the public, in that scenario, the client would be an individual.

Here are a few scenarios that may help

ABC Law Firm (Company) has 25 Attorneys (Contacts) that work for them.  

ACME Insurance (Company) has 5 Adjusters (Contacts) that work in the office.  ACME is the company you will enter in the system one time.  The 5 adjusters would be entered individually as contacts for that client.

For more information on adding and managing contacts, check out the video tutorial below