Puisque has a cause/effect meaning and it is used when since is a synonym of because:

Since you are in the kitchen, could you bring me a cup of coffee?
Puisque tu es dans la cuisine, pourrais-tu m’apporter une tasse de café ?

Depuis is for the cases where since have a meaning of time, space or when we are speaking about series:

Time:

This company is serving customers since 1892.
Cette société est au service des ses clients depuis 1892.

Space:

You have been bothering me since Paris!
Tu m’ennuies depuis Paris !

Serie:

They are all there, from the first to the last.
Ils sont tous là, depuis le premier jusqu’au dernier.

Mind that this one could also, in a less formal way translate to

Ils sont tous là, du premier au dernier.