Well, I don't know how useful or accurate this info will be but here goes. That being said, you can find all this info and more in better detail with my main man, google.
Recommended reading/watching:
-Please Kill Me- Legs Mcneil (influential figure) This deals with the New York side of things, but is equally important as far as the history of punk.
-Westway to the World: A great documentary on The Clash. They have quite a few docs to watch but this one was great.
-A million books written about each band, the social issues of the time, and punk music at large
Background The popular music of the early 1970's was, in general, musical excess. With electric/space focused music utilizing synths, long, overly complex guitar solos, it was the absolute opposite music compared to the proto-punk or early forms of punk music originating in places like New York or London.
England specifically catered to the musical genre of punk rock. A record high for unemployment (especially for youths and young adults) and a growing division of wealth between white collar and the working class poor led to the political vacuum of the era.
Most of the members of The Clash (Joe Strummer, Mick Jones, Paul Simonon) met in art school (or around that time) and had a working knowledge of music and how to play instruments. The Sex Pistols were blue-collar as far as background and upbringing, which kinda shows in their musicianship and lyrics.
Differences The biggest differences between these bands was there actual music. The Sex Pistols, while being the "face" of punk music of the era were far from being the driving force. Sid Viscous, their notorious bassist joined fairly late in the bands' career and by the time of his death, couldn't really play the bass. The Pistol's only recorded one real record (aside from b-sides and demos), though a genuine part of musical history. Most of the lyrical content of the Sex Pistols reflected their blue collar lives, while the Clashes' lyrics were more complex, showing their college education. The Clash also incorporated different genres of music into their music. Reggae music, rockabilly, it all had a punk edge to it when the Clash picked up on it. Both bands were socially liberal (with the exception of the Pistol's "Bodies" about abortion), each band took a different approach to music and how they marketed themselves. Both bands preached about income inequality and anti-establishment views (while making deals with major record companies), both bands generally maintained a DIY ethos.
Impact The impact of both bands is still seen today. Both bands still are played on popular radio, if not "classic rock stations," though the Clash seem to get more airplay due to having a larger discography to play from. Posters, shirts, and merch (Irony) can be found in malls all over the world, and punk aesthetic and sub-culture has long bled into mainstream pop-culture. *Even yesterday, while buying some stuff at a craft store, London Calling came on the radio as I was paying for my stuff. Critics could say that the Clash come off as more of a "poppy" sound compared to the Pistol's, however, I believe that is a skewed way to compare them. First, as mentioned above, The Clash, FOR THE MOST PART, could play their instruments. They made songs that, simply put, SOUND better if not for a general sense of musical cohesiveness. Their music was, from a musical standpoint, more complex, whether dealing with lyrics or theory.
My stupid opinions My own opinions on the subject- Listening to both bands, the differences should be fairly obvious to even someone unfamiliar with the bands or punk in general. Though I have respect for The Sex Pistols and their coveted place in history, The Clash are easily the better band. More tunes, far better lyrics, and music that still holds weight today.
This is all from memory, so ALL of it can be expanded on. I hope it doesn't come across as a hit-piece on The Sex Pistols, for they do deserve their rightful play in history, but I and many of folks agree that the Clash have longevity, the Pistol's had angst. Go check out wikipedia to fully flesh this stuff out, and there are many engaging documentaries and books about these amazing bands.
Anyone fell free to expand on this, or tell me exactly how and why I'm wrong, PUNK BIIIIATCH
DEATH OR GLORY