They mean more or less the same thing.
'Light' suggests something that's fun and less serious, whereas 'bright' suggests optimism and a positive outlook.
If you tell someone to 'Look on the bright side' , you're telling them to consider the positive aspects of an otherwise negative situation.
The expressions with 'lighter side' are not very common, but suggest a light-hearted (joking) attitude. For example, the title of a magazine column about somebody's funny experiences at work might be 'John Brown looks at the lighter side of ......(whatever work he does) .......... ' . 'Look' means 'consider', whereas 'see' means 'be aware of'. So if you can 'see the lighter side of something', you can appreciate what is also funny about a serious, or neutral situation.
I hope that's clear for you.