Both the tufte-handout class and beamerarticle.sty load the xcolor package and with different options, that's why the error message Option clash appears.
Specifically, tufte-handout.cls uses tufte-common.sty and this requires xcolor by this line:
It allows you to use xcolor features with the tufte classes. But as soon as you load beamerarticle it would try to pass again an option to xcolor. But beamerarticle is cleverly programmed and offers the option
noxcolor which simply doesn't load xcolor, making it compatible for such cases.
% For color support