Evoke means (1) to summon or call forth, (2) to call to mind, and (3) to call up a memory from the past. Invoke means, primarily, to call upon, especially in reference to aid, assistance, or a higher power. Less common definitions include to cite for justification (as, for example, when a lawyer invokes a precedent to make an argument), to conjure, and to resort to.
The words are similar, and in some senses they can be interchangeable. But think of it this way: Invocation involves making a plea, and evocation involves summoning something.
Examples
These writers use evoke well:
As with terrorism, the public has deep concerns about America’s place in the world, but these worries do not evoke a strong policy debate. [The National Interest]
It’s intended to evoke a sense of nostalgia … [Miami New Times]
The Irish Times went so far as to evoke the memory of WB Yeats in its unnerving editorial “Was it for this” two weeks ago. [Guardian]
Some writers use provoke (which connotes intention to evoke or elicit, often with sinister goals) where evoke would make more sense—for example:
CCTV cameras provide reassurance to some and provoke unease about a surveillance state in others. [Herald Scotland]
A new tribute to Mark Twain may provoke as much thought and provide as much enjoyment as the author’s words. [Hannibal Courier-Post]
And these writers use invoke well:
One is reminded of Dante, who invoked the muse to speak of his journey to the pit and back … [James Blachly]
Desperate for money, the city of Portland decided to invoke a leaf-removal fee this autumn. [The Oregon Commentator]
Danes still sometimes invoke Tycho when they explain their need to excuse themselves during a meal. [NY Times]

