widen and broaden

The two words are very similar in meaning. Sometimes they can be used interchangeably. In some situations, one is preferred to the other.

widen : to make something larger in degree or range. Widen is about the quantitative change of something.
e.g.
widen the entrance (physical object)
The gap between the rich and the poor is widened.
widen the scope of activities
The constituency is widened to include more voters.

broaden: Besides having the meaning of making something wider, broaden is used to describe an increase in experience or knowledge. In short, broaden is used for something not easily quantified.
e.g.
broaden the road (physical object)
broaden the access the higher education
The party tries to broaden its appeal to voters by compromising on principles.
Travel broadens the mind.
Doing volunteer work broadens your horizon.

If you could say : "under broad day light" and not "under wide day light", I guess broaden could sometimes means: clear/ not hidding something.

Example :

1) John has broaden his understanding of this difficult subject.
I would read it as : John has clearer understanding on the confusing subject.

2) John has widen his understanding of this difficult subject.
I would read it as : John has further misunderstood the difficult subject.

A good question...
No difference