Most obsidian will not allow light to pass through. Apache tears will, though. This kind of obsidian appears opaque until you hold it up to light, and then you can see it is translucent. There is also black obsidian (easily mistaken for jet on sight alone), snowflake obsidian, mahogany obsidian, and different colors of sheen obsidian, which shimmers when you turn it in the light.
Obsidian is volcanic glass. A lot of times it is tumbled with a shiny finish, which I have never seen on a piece of jet. I have a tumbled piece with a matte finish which does look a lot like jet, but if you feel it, its weight and texture are glasslike -- kind of like weathered sea glass or those matte glass marble discs used in flower arranging and such.
Jet is fossilized coal, and is essentially a form of petrified wood. My pieces are lighter in weight and have a more woody feel to them than the obsidian. They are all matte, and once in a while you may be able to see wood grain if you look very closely.
It is easy to learn to tell these two stones apart if you can spend some time with a piece of each and really look at and feel each one. They do look a lot alike, but they feel very different.
-- Kyrielle
