I'll indulge, having recently rewatched the series and reread it within the last year.
How would the story be different if Cersei married Ned?
The point of such speculation is usually to arrive back at a story line which somewhat resembles the original. Otherwise what is the point, as we end up discussing nothing which resembles the original work.
" What would have led to this union? "
Well in order to "highjack" the plot, such a union would have had to predate his marriage to a Tully. So either Ned would have had to have wed Cersei prior to his brother's death or Ned's brother would have had to been betrothed to him instead of Catelyn. Either way, it is relatively easy to manufacture such a scenario, largely because very little is known at all about the motivations of Ned's parents. In fact, I just went and looked up information about his mother, because she goes virtually unmentioned in both series, except one vague mention of her having a Flint in her ancestry. Turns out Ned's Dad married a 2nd cousin, once removed. I think we can accept it as canon, as Martin has said he uses the wiki himself. (Turns out her mother was named Arya Flint, which is interesting in itself for what this says about possible scenarios in the books and maybe the tv show. The same goes for my discovery about the connection of the Starks to the Royce family and how it explains Ned's own fostering.) So, why would they be married? Well they are both great houses. The Lannisters are richer, but are considered "new money" as is shown by the Mad King refusing to consider a marriage between his son and his "servant's" daughter. So the Starks might have provided the unimpeachable ancestry, that Tywin might have sought after being thrown aside by the king. The rich have always married "up" into the aristocracy and such a match would have appeared to be a good one for a second son such as Ned, as since Jamie had become a Kingsguard and Tyrion was a dwarf, this means that Ned's children would have inherited Casterly Rock. I think that is an acceptable reckoning of how it would come about, as it is never firmly established how the original marriage was arranged, other than some hints scattered among stories and flashbacks of when they were younger. There is an implication of something of a love match and that Ned's brother was well known at Riverun, but it isn't really explained much.
"Would Robert's Rebellion still be successful",
Well, the basis of the rebellion has always been the Stormlands, the Aerie, the North, and the Riverlands. The Lannisters are credited with the final blow, when Tywin betrays the king and Jamie Lannister slays him. The riverlands only joined because of the two marriages, so I don't think that one marriage to House Arryn would have sufficed. Now the easiest "swap" I can think of would be to posit Robert marrying Catelyn (You can go think about this now, I've returned the favor, I find this even more interesting an idea than yours) but this isn't really possible, given that he still hoped to rescue Lyssa at this point. So lets pretend that the hope of such a marriage and the actual marriage to Jon Arryn would suffice. That leaves the only real questions being about Jamie living and King's Landing being unburnt. I think we can assume that Tywin Lannister would have played the fence up to the last moment still and pulled it off in some manner.
How "close" would they be?
Now within the tv series, Cersei's motivations are spelled out explicitly as being a reaction to Robert's behaviors. Sure she and Jamie had diddled around some, but they make a point that Cersei was once idealistic and was as silly as a little girl about the romance of her marriage. That she was changed by the death of her and Robert's only child and his behavior in the wake of that pregnancy, birth, and death. Her motivations are driven by Robert's behaviors, his drinking and whoring. So my first reaction is to say "Why she'd never have turned out the same if married to the honorable Ned Stark, a sober and conscientious man." Yet Ned too had a bastard.
"What could this mean for Jon Snow?"
Most likely death or an early exile and being raised like Ramsey Snow. One of the interesting dynamics brought out in the tv series is the relationship between Jon and Catelyn. Her story of her hate for Jon being thrust in her face and her terrible guilt over that hate when he fell sick is one of the more interesting parts of her character, a deliberate way to contrast her and Cersei's deep devotion to their children. They'll both do anything to protect their children and secure their places in the world, but Catelyn has a conscious. She at least feels regret that she can not love Ned's son, the one who is always "sulking in the corner." So I imagine that Cersei would not have stood him at all. He'd have been in the stables, and she'd likely have tried to kill him at some point, while Ned would steadfastedly refuse to send his sister's secret child away. Yet. Look at Jon's fate. Ned knows King's Landing is not the place for his "son" and when Catelyn refuses to allow him to remain in Winterfell without Ned, he agrees to the subtle manipulation of Jon which has led him to ask to join the Night's Watch.
"she and Jaime have the same relationship? "
No. Jamie would have remained in the Kingsguard or returned to Casterly Rock, given the only reason he remained in the Guard and at Court was to be around Cersei. There is really no good explanation for how he would have ended up at Winterfelll as anything but an occasional visitor, so any secret meetings between him and Cersei would have been far apart, In addition, Winterfell isn't the Court and Ned isn't a drunken whoremaster, so the chances of any affair going unnoticed would be negligible. In addition, Ned isn't likely to allow his wife to spoil his children ( or are they blaming Joffrey on inbreeding?) and he wouldn't have been an easy cuckold.
The only way I can arrive at the incest twins is to bring up an alternate idea once advanced, that the only thing which kept Ned Stark from sitting on the Iron throne after he told Jamie to get off it was that he didn't want to be king. That his friend Robert would have just as happily supported him as king as be king himself. For that matter, Tywin would still want to put his daughter and grandchildren on the throne.
Now that does make for an interesting idea, as it would allow Jamie that access. You make Benjen, Lord of Winterfell and the King's brother instead of Stannis and Renly, Maybe that is who Catelyn marries after Brandon's death Jon Arryn would likely still be hand, but to a king who wanted to rule well.