Carter strikes me as an overall poor choice for VP, someone like Fritz Hollings or Sam Nunn might be better southern choices. Lloyd Bentsen is another. John Glenn also wouldn't be a horrible choice and would be a fresh enough face in the Senate to avoid having too much "insider" baggage.
But okay, let's say it's President Jerry Brown and Vice President Jimmy Carter. I personally foresee Pres. Brown's "era of limits" attitude clashing with Congressional leadership in ways not entirely dissimilar from Pres. Carter of OTL. Brown supported a Balanced Budget Amendment, opposed universal healthcare, generally slashed government spending in ways big and small as Governor of California. He was quite miserly at this time, and he was unafraid to call out what he considered wasteful spending. He declined to live in the governor's residence and instead rented an apartment in Sacramento, walked to work or drove himself in a modest car as opposed to taking the more traditional limousine. Pulled right from Wikipedia,
"When Gray Davis, who was chief of staff to Governor Brown, suggested that a hole in the rug in the governor's office be fixed, Brown responded: 'That hole will save the state at least $500 million, because legislators cannot come down and pound on my desk demanding lots of money for their pet programs while looking at a hole in my rug!'"
This kind of attitude is going to ruffle feathers in Washington. Where Carter was dismissive of Congress, and as many have suggested in the past, somewhat ignorant of how the President should relate to Congress, I think Brown might be more overtly combative. Maybe not; I don't know how he handled the state legislature during his time as Governor, so maybe he knew a thing or two about finesse. Another interesting area where I see potential for similarities is in Brown's staunch environmentalism. President Carter famously tried to convince the American people to conserve their energy use, even making that public address where he urged them to "put on a sweater" instead of turning up the heat in the winter. It's my understanding that initially it was actually received well, but public opinion turned against it ultimately. I wouldn't be surprised if Brown did the same thing, though the outcome could be more positive for him. After Three Mile Island in 1979, he became an opponent of nuclear energy, but I don't know where he stood on it before this incident; he might support it as President only to do a 180 if the same (or a similar) event occurs here. I think he would certainly support solar energy and probably also have solar panels installed on the White House roof, like Carter did.
Related to the energy issue is of course relations with Iran and the Iranian Revolution. I don't know much about his foreign policy planks, and as Governor he wouldn't have much experience there (all the more reason that choosing Carter for VP might not be wise) but I will say that based on what I know about the man, I think
@GeographyDude is onto something. President Brown seems like he might just be bold enough to bend the knee and take a more noble course towards Iran - and I think he would be less than overly sympathetic to the Shah, avoiding the blunders made by Carter which inflamed tensions between Iranian revolutionaries and the US, helping lead us to the hostage crisis. That alone would be very good for the trajectory of his presidency.
President Brown might not see the total collapse of popularity that President Carter did OTL. In a lot of ways I think he would be more effective. But as I have said, I do think there would be some broad similarities, and I do subscribe to the idea that 1976 was actually something of a poisoned chalice. I could see big potential for Brown's tenure to look very similar to Carter's and if he fumbles really badly, tension between himself and Ted Kennedy might just lead to a primary challenge like OTL. This seems like a worst-case scenario though, and I think it's somewhat more likely that Brown does lose reelection in 1980 but not in a massive landslide. Unless he alienates all sides of the spectrum, which does not sound at all unlikely either...