Something that is complex can be broken down into simple parts, without being simple itself. This allows mostly smooth functioning with solvable problems.
Something that is complicated cannot be broken down into simple parts, for it is made of complicated parts. “Complicated” is indicative of something that is problematic, convoluted, difficult and inconsistent.
The difference can be made by imagining a piece of string tied to be a circle. Now imagine using that string to play cat’s cradle or another of those childhood string games. If done successfully, one will weave a web of complex structure that can be followed from one side of the knot all the way through the string to the other side of the knot. While it bobs, weaves, and often looks confusing and difficult at first glance, it can be done. When the game gets botched, however, and one ends up with a tangled mass of string- that is complicated. One cannot follow the string end to end because it is virtually impossible without first at least partially untangling the string. Like the complex version, it bobs and weaves and looks difficult at first glance. It also looks difficult at second and third glances.