It was known as The Cousins War. While the English Civil was a war between English institutions; the Crown and Parliament, the War of the Roses was a war between an extended family.

More than anything, it's the contrast in scales and impacts of the conflicts. In proportion to the population, as many people died in the Civil War as in WW1. The war also fundamentally changed the political structure of the country in a pretty unique way. These large scale social and political impacts are what justifies it being called the English Civil War. Although recent historians have revised their attitudes to the Wars of the Roses to a more important impact on English society, it has traditionally been seen to have had a minimal effects outside the political elite, with next to no impact on social or political structure, other than the ascendency of the Tudor dynasty.

It's important to note that the War of the Roses was far from the first war to divide England.

There was The Anarchy from 1135-54, which saw the end of the House of Normandy and the rise of the Plantaganets. One could perhaps argue that this was actually the first English civil war, at least since the Norman Conquest.

Then there was the First Barons' War from 1215-17, which saw rebellious barons backed by French and Scottish forces trying to overthrow King John.

Then there was the Second Barons' War from 1264-67, in which Simon de Montfort rebelled against Henry III, only to be killed in battle later on against Edward Longshanks (known popularly as the baddie in Braveheart).

Then there was the Despenser War from 1321-22 between the lords of the Welsh Marches and Edward II.

There could well be even more than that.

In terms of where the Roses thing came from, the war was between two families (really sections of the same family, the Houses of Lancaster and York), each of which had a rose as their symbol; one red and one white. When Henry VII took the throne, he combined the two to create the Tudor Rose.

As for the Civil War thing, pimasecede is right; the war was between two family factions, it didn't involve the government or any institutions.