Etymology

The term wrestling is an Old English word that originated some time before 1100 A.D. It is perhaps the oldest word still in use in the English language to describe hand-to-hand combat. The Merriam-Webster online dictionary defines wrestling as "a sport or contest in which two unarmed individuals struggle hand-to-hand with each attempting to subdue or unbalance the other".
Roget's New Millennium Thesaurus does not support the usage of 'wrestling' (noun) and 'grappling' (noun) as synonymous.

Wrestling disciplines defined by FILA, are broken down into two categories; International wrestling disciplines and folk wrestling disciplines. According to the International Federation of Associated Wrestling Styles, there are five current International wrestling disciplines acknowledged throughout the world. They are Greco-Roman Wrestling, Freestyle Wrestling, Grappling, Beach wrestling and Sambo.

Greco-Roman
Greco-Roman is an international discipline and an Olympic sport. "In Greco-Roman style, it is forbidden to hold the opponent below the belt, to make trips, and to actively use the legs in the execution of any action." Recent rule changes in Greco-Roman increase opportunities for and place greater emphasis on explosive, 'high amplitude' throws. One of the most well known Greco-Roman wrestlers is Alexander Karelin from Russia.

Grappling
"Grappling is a wrestling style also called submission wrestling or “submission grappling” which consists of controlling the opponent without using striking, in standing position or on the ground after a throw, and to make him or her abandon thanks to immobilisation techniques such as locks. Grappling plays an important role in the practice of Mixed Martial Arts (MMA) and can be used as a self-defence technique. It brings together techniques from Brazilian jiu-jitsu (no-Gi), Freestyle Wrestling, Folk American Wrestling (catch-as-catch-can), Sambo and judo.".[17] Grappling is also used to describe the skills used in mixed martial arts competitions, differing from the FILA definition. Grappling can be trained for self defense, sport and mixed martial arts (MMA) competition.