Both fürchten and befürchten mean "to be afraid" in some sense, and although it's a bit confusing at first, have no fear! Let's first take a look at some of the differences, starting with fürchten:

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Es war richtig gewesen, den Wolf zu fürchten,

It had been right to fear the Wolf,

und richtig sich deshalb ein Steinhaus zu bauen.

and therefore right to build a stone house.

Captions 80-81, Märchen - Sagenhaft - Die drei kleinen Schweinchen

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Denn ihn fürchten sie: den Adler.

For it is the one they fear: the eagle.

Caption 19, Die letzten Paradiese - Die Schönheit der Alpen 2

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The verb fürchten is usually translated as "to fear" and can be used to describe the state of having real fear, of being truly afraid of something like a dangerous animal. But it can also be used to show fear of a situation: 

Viele Demonstranten fürchten,

Many demonstrators fear

dass Minderheiten in den USA nun kein Gehör mehr finden.

that minorities in the USA will now no longer be heard.

Captions 41-42, Tagesschau - Amtseinführung von Donald Trump

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The verb fürchten is also often used reflexively:

Ich trag' mein Licht und fürcht' mich nicht...

I'm carrying my light and have no fear...

Caption 3, Sankt Martin - Das Laternenlied

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Therefore, ich fürchte mich is another way of saying ich habe Angst or "I am afraid," and afraid in a serious way.

The verb befürchten, on the other hand, is never used reflexively and is generally used in situations where you're not literally afraid or having serious fears, but rather in situations where you are merely worried about something.

Sie befürchtete, dass ihr Herr noch immer in Gefahr war.

She was afraid that her lord was still in danger.

Caption 71, Märchen - Sagenhaft - Ali Baba und die 40 Räuber

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This could also have been translated that "she was worried" about her lord. 

Es ist, wie ich befürchtet hatte.

It is as I had feared.

Caption 10, Oskar - Gehen, wenn es am schönsten ist - Nur vier Wochen

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So even though befürchten is translated here as "to fear," it could equally have been translated as "to expect" with the implied connotation of expecting a negative development. 

Unfortunately, sometimes even native German speakers use fürchten (to fear) as a way of exaggerating a circumstance where befürchten (to be worried about) would be more appropriate, and also vice versa. The German Duden dictionary, the standard for the German language, is quite clear on its definitions of the two words, however. To reiterate: fürchten should generally be used to express real, direct fears, and befürchten in milder situations to express worry.

Further Learning
Translate these Duden definitions of fürchten and befürchten, then go to Yabla German and find some more examples of videos using these verbs. 

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fürchten:

1. vor jemandem, etwas Angst haben; Unangenehmes ahnen, befürchten

2. Furcht empfinden, Angst haben

3. sich vor der Prüfung fürchten

4. vor jemandem Ehrfurcht haben

befürchten (only one definition):

(etwas Unangenehmes, was vielleicht eintrete könnte) aufgrund bestimmter Anzeichen oder intuitiv erwarten, kommen sehen.