Evoke and invoke are close together in meaning, and are often confused with each other.
The action of invoking is usually more direct and active. It originally involved calling upon or summoning up a god or spirit. An invocation calls upon whatever is invoked to do something or serve a function. Invoke now can also be used to mean to appeal to, to cite: in his closing argument, the lawyer invoked the principle of self-defense.
Evoke is usually less purposefully active, more indirect, often used to mean suggest. If you invoke the spirit of Picasso, youre trying to summon his soul up from the grave; but if your paintings evoke the spirit of Picasso, it means their style reminds viewers of that artists work.