In the spirit of Paul Cantor's Gilligan Unbound, I thought it would be fun to compare the (original) Star Trek with Joss Whedon's Firefly with the aim of thinking about how popular culture has radically shifted over the past 30 years. Unfortunately my html skills aren't up to making a side by side look, so you'll have to bear with my presentation. For the benefit of any of you who haven't seen Firefly, I'll mention quickly that like Star Trek, it is a set of stories about a space ship which travels from planet to planet, though Firefly has more of a Western feel to it instead of being a traditional sci-fi show.
Star Trek: to explore the galaxy in servive to the galactic federation, with the typical result being the use of force to overcome the local way of doing things, unmask religious pretensions, and often use overwhelming force to bring about defeat or conformity to liberal/democratic priciples (you'll need to read Cantor's book for a full discussion of all of that if you are skeptical).
Firefly: Typical aim is some sort of short term money making enterprise where the members of the ship need to use all their cunning to escape in one piece.
ST: an ethnically diverse group of staff members all serving to further the overall aim of the federation. Only three of these play a significant role in developing most of the plot lines.
Ff: a diverse group made up partly of staff relatively loyal to the captain, but also including some who have ended up on the ship by accident. Each character on the ship plays a part in the stories and each has a separate way of looking at their position onboard.
ST: The Enterprise is a state of the art warship, equipped with the highest levels of weaponry available. These weapons often come into use in the stories. The characters are personally armed with sophisticated weapons and other medical and communication devices, relying heavily on these technologies to acheive their aims.
Ff: The ship is an obsolete model, completely devoid of weapons. While some of the characters will use firearms, these are generally old fashioned guns. Because of their overall powerlessness, the characters rely on wit and trickery and bravery rather than technology for the most part.
ST: The Federation is viewed as a benevolent force trying to civilize the galaxy. The cast serve the federation generally without question and do not dispute it's views of governance.
Ff: The Alliance is generally portrayed as a poweful enemy to the cast. Its views of governance are hidden but assumed to be corrupt and insidious. The ship tries to avoid and work around the influence of this government.
ST: Cpt Kirk is a tradition naval commander, expecting total and immediate obedience to his commands. All decisions are ultimately his and he only allows hs senior officers to question him or discuss decisions.
Ff: Captain Reynolds is much more like the captain of a commercial ship. He is employer to some of the crew but others have a more ambiguous relationship with him. One character rents space on the ship and expect the captain not to enter her shuttle without permission.
I could go on, but perhaps you would like to add some of your own.