The FO4 Brotherhood are the same as the FO3 ones, but they've developed into a new branch of Brotherhood under Maxson, much like they had under Lyons. They follow a more militarised command structure (started by Lyons) and have more power centralised in a single Elder when compared to the west coast Brotherhood.
Maxson's generally dickish behaviour comes as a result of being raised in the Capitol Wasteland, which was more of a mess than the NCR region they lived in on the west coast. He was appointed to be leader after a number of ineffectual Elders kept making bad decisions while trying to keep to Lyons' philosophy of helping people and being guardians of the wasteland settlements. Maxson basically said 'screw that shit' and decided that it was more practical to focus on scouring the wasteland of mutants, feral ghouls and other such threats. He also managed to get the more traditional Outcasts back into the fold, which probably affected the rest of the Brotherhood's outlook on things.
Since the Brotherhood had been struggling under previous Elders they took to this new more authoritarian rhetoric promising change like ducks to water. There's an element of historical revisionism among the Brotherhood in FO4, some of the NPCs comment in passing that Lyons was a weak and bad leader, even though he was quite popular, if controversial, among them in FO3.
The Brotherhood of FO4 are different from the other versions in that they're basically an authoritarian military junta lead by a demagogue. They do care about civilian life to a certain extent, but they place their own needs and goals above the sanctity of human life. Compare to the west coast Brotherhood who cared about people, protecting technology from the ravages of time and were willing to make deals with other groups, and were lead by a council of the oldest members, and to the FO3 east coast Brotherhood who placed protecting people above technology and had a largely civilian leadership headed by their oldest member who held his rank as a result of the traditional power structure they had left behind.