What exactly is the difference between 'It evokes our fear' and 'It provokes our fear'? Is the difference really huge?
A huge difference? Probably not in this example.

But here are a few things that I see as being different:

I think you are provoked with deliberate actions, and evoked is more passive. A smell of baking bread can evoke memories of your grandmother's kitchen. You could never use provoke for that.

Provoke is almost always used with negative emotions or actions, so while it applies to fear, perhaps, it would not apply to happy memories.

I also feel like when you are provoked, you are called to some sort of action, while something that is evoked is called up within you, particularly emotions.

I was provoked to punch him. His words evoked my anger.
evoke: it calls the fear from within us, brings it forth
provoke: causes, gives reason for the fear

This is a little vague, but I hope it helps.
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Great answers! Thank you both! 
According to englichforums.com ....

The answers from Princeton:

# arouse: call forth (emotions,
feelings, and responses); "arouse pity"; "raise a smile"; "evoke sympathy"

# arouse: call forth (emotions, feelings, and responses); "arouse pity"; "raise a smile"; "evoke sympathy"

# raise: summon into action or bring into existence, often as if
by magic; "raise the specter of unemployment"; "he conjured wild birds in the
air"; "call down the spirits from the mountain"
# appeal:
request earnestly (something from somebody); ask for aid or protection; "appeal
to somebody for help"; "Invoke God in times of trouble"


apparently, evoke and provoke are synonymous while invoke carries
a slightly different meaning.

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