Focus on a single time period
Part I and III were primarily focused on a single time period: 1955 and 1885, respectively, with about ¾ of each film taking place in their respective time periods. In part II, ¼ of the film was set in 2015, ¼ in 1985, and ½ in 1955.
Following Marty through the DeLorean
Part II is the only film where we see Marty and Doc travel through time together, with the exception that we see them travel together at the end of part I. However, we do not follow Doc and Marty in part I, as the film ends after they leave 1985. It is not until the beginning of part II where we see them depart together and then arrive again 2015. Early in part I, we follow Marty through the DeLorean as he travels from 1985 to 1955. Toward the end of the film, we follow Marty again from 1955 back to 1985. Early in part III, we see Marty travel from 1955 to 1885, and toward the end of the film, from 1885 to 1985. However, we do not follow Doc and Marty time-travel together from one time to another in parts I and III.
Saving Doc's life
Early in part I, we see Doc get shot and killed by terrorists, shortly before Marty travels to 1955. Early in part III, we find Doc's tombstone dated September 7, 1885, which is why Marty traveled to 1885 instead of back to 1985. Though the main plot of part I is about how Marty disrupted the events that led to his parents falling in love, and attempting to get them back together so Marty can have a chance to be born, we do see Marty attempting to warn Doc about his death in 1985. In part III, Marty has to warn Doc about his death again by showing him a photograph of his tombstone. However, Marty does not have to save Doc's life in part II.
Marty's alias
In both parts I and III, Marty uses a celebrity's name as to not reveal he has the same last name as his father (in part I) or great-great grandparents (in part III). In part I, he uses the name Calvin Klein, and in part III, Clint Eastwood. Though we do hear Lorraine refer to Marty as Calvin Klein in part II, he does not come up with a new alias in part II.
Leaving at will
In parts I and III the DeLorean can only be used at specific times, because of either a lack of plutonium (part I) or gasoline (part III). In part I, Marty can only return to 1985 at the time lightning his the clocktower, on November 12, 1955 at 10:04pm, since a bolt of lightning is the only way they could power the time circuits. In part III, Doc and Marty can only use the DeLorean when the next train arrives, on September 7, 1885 at 8:00am. Though they have Mr. Fusion to power the time circuits in part III, they don't need plutonium. However, Mr. Fusion only powers the time circuits and not the DeLorean's engine, so without gasoline they cannot get the DeLorean to 88 miles per hour. They need the train to push the DeLorean to the required speed so they can travel back to the future.
In part II, there are no barriers preventing them from using the DeLorean when they finish their mission in the given time period. In 2015, this is after they saved Marty Jr. from participting in a robbery with Griff, and in 1985 when they learn when older Biff gave his younger self the sports almanac.
Naming Places
In part I, Doc and Marty meet at the Twin Pines Mall. After Marty arrives in 1955, he drives over one of the younger pine trees. After arriving back in 1985, he goes back to the mall's parking lot which in the altered timeline has been named the Lone Pine Mall.
In part III, Doc saves Clara Clayton's life by stopping her from falling into a ravine and a little bit later, Marty tells Doc a story about how a teacher fell into the Clayton Ravine, which is how the ravine got its name. Later in the film, a locomotive pushes the DeLorean to 88 miles per hour and sends Marty back to 1985, then the locomotive falls into the ravine. Though Doc and Clara know that Marty was not killed but was safely returned back the future, as far as anyone else in 1885 knows, Marty fell in the ravine. Since Marty introduced himself as Clint Eastwood in 1885, the ravine was renamed the Eastwood Ravine.