Whereas & despite

They are used to compare elements. 

Whereas is used to compare two contrasting ideas in the same sentence (clause+whereas+clause)

Many soldiers supported the war, whereas others protested against it.

The government defended the invasion, whereas the population opposed it.

Despite/In spite of shows contrast between expectation and reality

Despite the danger, soldiers kept fighting until the end

Despite public protests, the war went on.

Despite his injuries, he went back to the battlefield

To sum it up, whereas shows contrast and despite shows unexpected result.

Whereas compares two clauses and despite introduces a noun or -ing form. 

Practice: 

Exercise 1. Choose the right word.

Many Americans opposed the Vietnam War, __________ the government increased military action.

__________ the protests, the president refused to withdraw the troops.

The war was officially justified, __________ many citizens considered it unnecessary.

__________ strong international criticism, the government continued its intervention.

Exercise 2. Combine the sentences.

The war was officially declared over. Many soldiers still suffered from trauma.

The government promised peace. It secretly prepared for war.

There were massive protests. The invasion continued.

Some veterans felt proud. Others felt ashamed.