Nucky-Eli Feud
From the first season until he's forced to leave Atlantic City in S4E12, Eli's jealousy of Nucky and his delusional belief that he can do what Nucky does constantly results in him making foolish decisions. He insists on making a speech at the Celtic Dinner that nearly ends up in a riot between the Irish nationals and Irish-Americans in attendance, which Nucky ends up having to smooth over.
The contrast is perfectly displayed in S1E8: While Nucky is in Chicago determining who the NJ Republican delegation will support for the presidential nomination and making side deals with the future Attorney General, Eli's sitting in Nucky's office telling Halloran he can do what Nucky does while the usually packed waiting room is empty and the only person who came in seeking a favor was O'Neil, whose first words were that he forgot Nuck was in Chicago.
It's understandable that Eli was angry when Nucky temporarily removed him as Sheriff, but after the casino shooting it had to be done so they could win the election, and unlike the other crime bosses who had to use violence to gain control of their territory, the only way Nucky could operate his criminal organization, have a monopoly on alcohol distribution throughout the city, and run all his rackets with impunity was if a puppet Republican mayor who he controlled won the election.
Nucky really summed up his jealousy perfectly in S1E11 when they had their first real argument and Nucky told him:
"You know it's too bad you didn't see Hardeen the other night...It's an entertaining act, but if he wasn't Houdini's brother, NOBODY'D GIVE A FUCK!"
Nucky almost single-handedly won the election by pandering to women's organizations like the Temperance League and League of Women Voters, the latter of whom endorsed Bader, by making a deal with Chalky for 100% of the Northside vote, and with his election eve press conference where he credited Eli's tireless work for the "apprehension" of the D'Alessio brothers. He gave him a portion of the $1M he got from Rothstein and within seconds of winning the election, handed him another full term as Sheriff, yet he still joined the "coup" that if successful would've landed Nucky years in prison so he could have a less important role in the political machine and answer to Jimmy and the Commodore rather than his own brother. When the Commodore had a stroke and he got angry that Nucky saw right through his bullshit apology, he suggested having him killed, then lied about it after the mistrial, which Nucky saw right through again.
In S4 he got angry and jealous that Will admired Nucky and went to him when he got arrested for murdering his classmate, and even with a gun to his head he complained that he went to Nucky instead of his father, but had he called Eli instead, the only thing he could've done was ask Nucky to intercede on his behalf because he didn't have any political connections who could get the case dismissed. Without Nucky's help, he would've sat in jail waiting to be tried for murder like his unfortunate roommate and never would've become an assistant US Attorney in the SDNY.
I think a big reason for their feud is rooted in their childhood where despite growing up in the same house with a kind and loving mother and a drunk and abusive father, Nucky was the one who suffered most of the abuse and because he took it on himself to work so that the family could have some of the basics, Eli's childhood wasn't nearly as rough as Nucky's. Their father obviously favored Eli, and burned Nucky's hand with a fireplace poker for taking a piece of bread and made him fight a group of older kids who beat him unconscious and put him in the hospital for 11 days. Eli gladly took his father in after he fell down in the kitchen and when they briefly met at the wake during S2, Eli couldn't understand why Nucky hated their father so much. Neither had it easy, but Nucky didn't have an older brother who started working as a kid to ensure they had food and Eli wasn't literally scarred from childhood because of his father's physical abuse. The Commodore was another asshole Nucky had to deal with for decades before he got to take over, and as demanding as Nucky was that his subordinates did their jobs, working under him was much better than working under the Commodore and everyone that was part of the machine did better financially under Nucky than the Commodore, even before Prohibition went into effect.
Eli got used to Nucky being the provider for the family and taking care of him, and rather than working hard on his own at something he was good at or taking his own shot at a profession he was suited for, he asked Nucky to get him a job from the political machine, but it wasn't until the Commodore had to step down as boss and serve a prison term that Eli was handed his first term as Sheriff. He didn't see Nucky giving him one of the best appointments or making him his #2 in the organization as things he should be grateful for; he expected that Nucky would take care of him and instead of knowing his place and his own limitations, he resented how popular Nucky was and that he'd always be viewed as his little brother rather than his own man.