Could someone explain how the usage of these two words differ?
You're probably talking about point 1a on Wiktionary.
Ich befürchte/fürchte, ich verliere meinen Job.
Both are fine and means the practically the same. You're using it like as a verb of saying in this one - indicated by the Verbzweitnebensatz "ich verliere meinen Job".
In any other usage, they are not similar at all. Please see the Wiktionary link for all its uses.
- fürchten can be used for people in two usages, but befürchten can not.
- fürchten is used reflexively in one usage, but befürchten is never used reflexively.
From my point of view, fürchten is a bit more universal, since it can directly refer to somebody or something (provided by an accusative object):
- Kinder fürchten die Dunkelheit.
- Ich fürchte mich.
Befürchten is typically applied to a situation, which follows e. g. by an auxiliary clause.
Ich befürchte, dass die Bankenkrise anhält.