They really should *not* be looked upon as the same band, although they were all part of a random evolution that took place over 20 years.
Jefferson Airplane is the undeniably great group. Jefferson Starship had its moments, but grew to be very crass and commercial (which some people admittedly like, but usually not Airplane fans). Starship represented a coup d'etat, when the remnants of Jefferson Starship were wrangled away from the last original Airplane member, and turned into a cynical cashcow, relying on outside songwriters. Nobody likes Starship except some aging housewife types.
Jefferson Airplane (1966-1973) was formed by Marty Balin, who assembled the pieces one by one. The original lineup was:
Maty Balin vocals
Signe Anderson vocals
Paul Kantner guitar, vocals
Jorma Kaukonen, lead guitar, vocals
Jack Casady, bass, vocals
Skip Spence, drums
After the first album, Skip Spence joined Moby Grape and was replaced by Spencer Dryden, and Signe Anderson quit, and was replaced by Grace Slick. This completed the "classic lineup", which flew from the end of 1966 to mid 1970.
The Airplane should be regarded has having two distinct (and frequently warring axis): The Kantner/Slick wing (vocals) and the Kaukonen/Casady wing (instrumental). Little by little the founder Marty Balin was odd man out, and quit in 1970. He was not replaced at first. Spencer Dryden was also nudged out and replaced with Joey Covington. Fiddler Papa John Creach was added in 1971, which altered the bands sound considerably.
This version of the band released 2 albums in 1971-1972 which are generally inferior to the band's classic side. A final incarnation of Jefferson Airplane existed in 1973, with David Frieberg added on vocals, to fill in the hole left by Balin's departure, and John Barbata replaced Covington on drums. This Airplane released one live album.
Essential to the identity of Jefferson Airplane was the presence of Kantner/Slick/Kaukonen/Casady.
Jefferson Starship came about in 1974 when Kantner and Slick wanted to tour again, and was told by Kaukonen and Casady "thanks, but no thanks". They decided to focus on their own band, Hot Tuna, which had begun releasing albums in 1970. Creach, Frieberg and Barbata from the late incarnation of the Airplane signed on, and Craig Chaquico came on as guitarist, and Pete Sears on bass. Marty Balin reappeared, on one cut on the first album, and as a full member on the next three, during which this lineup remained stable (except for Creah, who was dropped after two albums). This edition of the band was more like the Airplane than unlike it, but losing Kaukonen/Casady meant the band lost its blues connection, and the band increasingly went after a very un-Airplane-like soft-rock sound that got pretty mushy as the decade wore on.
In 1978, after four albums, Balin quit again, and Slick was fired (she would miss an album, and rejoin the band in 1981) Drummer Barbata was nearly killed in a car crash, and couldn't perform anymore. Kantner/Frieberg/Sears/Chaquico added Aynsley Dunbar on drums and Mickey Thomas on vocals. Thomas basically was intended as Slick's replacement.
Thomas and Chaquico would dominate this edition of the band, with Kantner and the returning Slick relegated to a more back seat role with each release, and Thomas singling lead on all the singles. The band's sound completely changed again, and became a hard-rock/arena-rock outfit like Foreigner. Generally, it is during this era that the spiritual connection to the Airplane was severed.
Dunbar was replaced on drums by Donny Baldwin in 1984, for what turned out to be the last Jefferson Starship album. Kantner, who had become resentful of the band largely being stolen away from him, eventually quit in disgust. As the only remaining original member of Jefferson Airplane, he sued the band to have "Jefferson" removed from the name.
So, the band became Starship. They promptly kicked Frieberg out before their first album and kicked Sears out before the second, becoming a unit of Thomas/Chaquico/Baldwin/Slick in 1986. Slick departed in 1988, leaving a trio of Thomas/Chaquico/Baldwin, none of whom had been in Jefferson Airplane, and only Chaquico had been in the original Jefferson Starship. So Starship really has nothing to do with the Airplane, except that it just kind of evolved into it. The remaining trio released one more album in 1989, and called it a day.
Musically, there is no comparison. The Airplane was bold, daring, literate, experimental, and capable of great beauty. Jefferson Starship was dull and syrupy, then was crass hard rock. Starship was one of the worst bands ever, pure adult contemporary crap.