Skipping rope is defined as follows http://www.thefreedictionary.com/skipping-rope
(Group Games / Games, other than specified) Brit a cord, usually having handles at each end, that is held in the hands and swung round and down so that the holder or others can jump over it
I've added a term to this n-gram to include jumping rope.
I've heard jump rope as the item used as well as the game, so that could weigh heavily since the usage is both a noun and verb. Skipping rope seems to be used the same way however.
Jump Rope is simply the term used in America and some other parts of the world used for the actual rope you jump when you are skipping . The term is now used to describe the activity as well. So particularly in the states you might jump rope using a jump rope!. The term is also used more now in the UK particularly because of the exchange of skills and equipment between the UK and the United States.
The use of the term skipping rope includes the activity as well in this example
Perhaps the usage of the terms viewed separately in American and British English will help as seen in these n-grams of the American usage here (where I added in jump roping just for fun) and the British usage here.
According to Wikipedia, the tool is called a jump rope or skipping rope, used for the game of skipping. As you can read in the article, there are many ways of jumping/skipping rope, and I grew up skipping in place over the rope in the USA as it was easier to keep going with less energy for competitions of duration rather than number of jumps.
Online Etymology Dictionary says about skip:
c.1300, "to spring lightly," also "to jump over," probably from O.N. skopa "to skip, run," from P.Gmc. *skupanan (cf. M.Swed. skuppa, dialectal Swed. skopa "to skip, leap"). Meaning "omit intervening parts" first recorded late 14c. Meaning "fail to attend" is from 1905. The noun is attested from mid-15c. The custom of skipping rope has been traced to 17c.; it was commonly done by boys as well as
girls until late 19c. Related: Skipped; skipping.
Notice that skipping rope has been traced to 17c.
Look here for a more recent history of the International Rope Skipping Federation (FISAC-IRSF)