What war was placed between the English and British?

Being English and being British are currently considered the more or less same thing.

However, in the fifth century, after the departure of the Romans from Britannia, there is the legend of war between the Britons (Welsh) defenders against the invading English (Anglo Saxons). The legend has the Britons under King Arthur finally defeating the Anglo Saxons and halting the invasion at the battle of Badon Hill in 495 AD.