Star Wars Purist said:
Star Trek never went into unadulterated, infant-focused products such as Episode I, II, and The Clone Wars. Yeah, there was TAS which was a Saturday morning cartoon, and the boring Motion Picture and elements of TNG and its successors that were also a bit boorish, and the new reboot was crap...but Trek has never had a "Jar-Jar" moment. In that sense, the franchise is "better".
However, as is acknowledged here, there really is NO COMPARISON of the franchises beyond their similar sounding "Star" titles, space-themed settings, and use of spaceships and high technology. etc. Star Wars is "Space-Fantasy" and Star Trek is Science-Fiction, yet both are, in their own ways, equally enjoyable and similarly faltering.
Personally, I think the Trek franchise is, in total, better, but nothing is as enjoyable, "fun", and influential (to my own daily life) as "Old" Star Wars.
To its eternal credit, TAS, despite being a Saturday morning cartoon, still did much to retain the intelligence and seriousness of TOS. It didn't have cute sidekicks added into the show, and the characters were all in-character.
Anyway, there really is no comparison. WARS is whiz-bang fun, TREK is intellectual drama, with a healthy dose of action and fun. Two sides of the same coin, which are perfectly capable of coexisting.
I'll try to compare them, though. It's more than a little unfair, because TOS had three seasons, a cartoon, and six movies to develop the characters and concepts.
Concept:
TREK: The Enterprise explores the vast reaches of the galaxy, encountering anything you could imagine.
WARS: A farmboy is swept up into a battle between the Rebels and Empire, and saves the day.
Winner: TREK, simply because the concept allows for limitless possibilities and stories, whereas WARS, in its original form, was a one-shot story with an ending.
Characters/actors:
TREK: Three fantastic and well-developed main characters, played by three great actors (and, yes, Shatner is great in TOS--this was before he became the Shatner we know today, after all). The secondary characters are mostly just bit players.
WARS: The characters are archetypal, and don't have a whole lot of depth, but that's not really the point. The performances are solid, for the most part.
Winner: TREK. You can't get more iconic than Kirk-Spock-McCoy. They're much easier characters to get a feel for, because we got to know them so well in so many episodes. TREK is more about the characters than WARS is.
Visual effects:
TREK: Cutting-edge for their time--movie-quality effects on a weekly TV budget. They only look bad now because better compositing techniques and motion control came along years after the show ended, and because TOS was never meant to be seen in HD on giant TV screens.
WARS: The films' effects revolutionized the industry. 'Nuff said!
Winner: WARS.
Production design:
TREK: The Enterprise, its interiors, the uniforms, props, etc. are all iconic, and very logical and futuristic. Setting aside the limits of 1960s materials, the designs themselves are timeless and unsurpassed by any other iteration.
WARS: Dazzling visual eye-candy, with designs drawing inspiration from many disparate sources, in such a way as to make the audience feel at home in an unfamiliar universe.
Winner: Draw!
Music:
TREK: Many TOS episodes have fantastic, powerful, and highly memorable scores created by great composers ("The Cage", "The Naked Time", "Amok Time", "The Doomsday Machine", etc.).
WARS: Williams' score reignited the popularity of orchestral scores in film, and the original movie's score has often been considered the best movie score of all time.
Winner: WARS
Impact:
TREK: Created the idea of modern sci-fi fandom as we know it. It's had more conventions, fanzines, and parodies than one could ever count. It launched the first movie series ever made because of fan demand. It's had a massive influence on real-world science, and has influenced countless artists and scientists.
WARS: Revolutionized the film industry, and cemented the idea of the blockbuster. A merchandizing empire unmatched in film history. A pop-culture juggernaut with a massive, cult-like fanbase which has led to "Jedi" becoming an official religion in some countries!
Winner: Draw!