'For the sake of' means, more or less, 'for the benefit of'. If you do something for the sake of someone, or something, they will benefit from your action. It is similar to 'for', but more precise in its meaning. For example.
'Although they wanted to get divorced, they stayed together for the sake of their children' .
This means that they stayed together so that the children wouldn't suffer.
'On behalf of' means 'in the place of' someone else. For example, if there is an award ceremony and the person who has won the award is unable to attend, another person (a relative or colleague) could go to the ceremony and receive the award on their behalf, or in other words 'instead of' them.
NB The expression 'on behalf of' often misused by native speakers. Many native speakers use 'on behalf of' when they should be using 'on the part of'. Do be aware of this.