What is the difference between binary and double stars?
They are not the same thing. Binary stars are actual physical systems, where the two stars revolve around each other. Double stars include binaries, but also include visual doubles, where two stars happen to lie along the same line of sight, but are not physically close to each other and don’t form a physical system. Variable stars vary in brightness. Some are close binary stars where one star eclipses the other, but most are intrinsic variables, where the star itself varies in brightness because of astrophysical processes going on in its atmosphere.
