Awarded –Rewarded

Please help me understand the difference.

1. He was awarded for his bravery.
2. He was rewarded for his bravery.
I see it this way:
1. He got an order or another decoration.
2. His bravery brought him some advantages.

Is it correct?

Les deux mots ont le même parcours étymologique : de l'ancien français (AF) vers le moyen-anglais.

An <reward>
AF <regard> <regarde> <regart> <resgard> <resgart> <resgar> <reguard> <regaurde> <revart> <reuard> <rewar> <reward> <rewarde> <rewart> <rouwart> <rewair> (ces étymons ont des sémantismes différents)

An <award>
AF <egard> <egart> <esgard> <esgarde> <esgart> <esguard> <esguart> <esward> <eswart> <eswar> <agar> <agard> <agarde> <aggard> <aggarde> <aguard> <award> <awart> (idem : sémantisme variable).

It looks like greg answered it. Please refer to his explanation.