Star Trek was first introduced as a television series going back to the mid and late 1960s. With the publication of novels, comics, animated series, toys and feature films, it grew into a full scale media franchise. Prior to that it was simply a television serial and known only as such that lasted only three seasons.
Star Wars was first introduced as a feature film in cinema going back to the late 1970s, when the first film, Star Wars Episode IV: A New Hope, came out in 1977; though a novelization based on the film's earlier script was published a year before the film. Star Wars quickly grew into a popular media franchise shortly after the release of the first two films and novels.
Star Trek has its origin in television and was only known as a television series. The franchise was conceived in the style of the TV western Wagon Train and the adventure stories of Horatio Hornblower but adopted a in the idealistic, utopian prospect of future human society. Star Treks main focus is giving a fictional depiction of space exploration and the system of a galactic society consisting of multiple planets and species. Conflict occasionally occurs.
Star Wars is an epic space opera was inspired by works such as Beowulf and King Arthur, and the origins of myth and world religions, Star Wars is a story that depicts a galactic society in constant conflict. Though there are periods of peace, this is not documented in the feature films but can be found in the comics and novels and spin-off films.
Aside from having the word 'Star' in their respective titles, the two franchises also share many similarities.
Both stories depict societies consisting of multiple planets and species. The main galaxy in Star Trek consists of various planets, both human and non-human, united into a single state known as the United Federation of Planets. Star Wars depicts a galaxy that is mostly part of a single state known as the Old Republic, inhabited by both humans and countless other species, that later became the Galactic Empire and later reformed into a new society called the New Republic.
Both franchises promote philosophical and political messages, though Star Wars not as much as Star Trek. The main philosophies of Star Trek convey the ethics of exploration and interference and how to morally deal with a new situation when faced by it. Creator Gene Roddenberry was inspired by stories like Gulliver's Travels that implied a morality tale. The main philosophical messages in Star Wars are the ethics of good against evil and how to distinguish one from the other. The philosophy of Star Wars also preaches against the totalitarian system and preaches in favor of societies that give equality to citizens.
The two franchises nonetheless have a "symbiotic relationship", states William Shatner, who credits Star Wars for the beginning of the Star Trek films. The documentary Trek Nation features interviews where both George Lucas and Gene Roddenberry praise one another's respective franchises, with the former stating that Star Trek was an influence while writing the original screenplay for Star Wars.
William Shatner was a presenter at George Lucas' AFI Lifetime Achievement Award ceremony. In 2011, Star Wars actress Carrie Fisher and Shatner posted a series of humorous online videos satirizing each other's franchises.
^Empire of Dreams: The Story of the Star Wars Trilogy. Star Wars Trilogy Box Set DVD documentary. 
^See David Alexander, "Star Trek Creator. The Authorized Biography of Gene Roddenberry" and interview with Roddenberry in "Something about the Author" by Gale Research Company and chapter 11 of "Trash Culture: Popular Culture and the Great Tradition" by Richard Keller Simon