There is a difference, however it's quite subtle. The way I would descibe it is as follows:

Stamina is associated with energy.

Endurance is associated with the ability to do something difficult for a long time.

For example:

A cyclist in the Tour de France needs stamina to climb the hills.

He needs endurance to climb hills every day for three weeks.

They are synonyms! However the word endurance is most often used to describe physical actions such as running long distances ex, "Wow, look at that marathoner go, he must have so much endurace!" Where as stamina is used often to describe other long, on-going mental actions such as staying awake for a very long time. Again, they are interchangable, so it really doesn't matter which situation they are used in. 

Hope that helped a bit :)

Synonyms! I feel really relieved! I'll just use 'endurance' for long distance sports. Thank you very much Amelia.