Here's my thought...
Sonic the Hedgehog and Jabba the Hutt.
Two brothers, young and old
Sonic is young: hedgehog lifespans are generally less than ten years; Sonic is depicted as a youthful character; and the first Sonic game debuted in 1991, so the franchise is only 26.
Jabba was roughly 600 years old by the time of Episode VI, making him quite old indeed.
Nothing in common, except for one word
The only thing they have in common is the word "the"!
One is family-friendly, the other is not
Sonic is aimed at kids.
Jabba is a ruthless gangster type who hires hit men and keeps slave women literally chained up by his side - when he's not feeding them to his pet rancor or to a Sarlacc. He's not particularly family friendly!
One is alive, the other is a memory
The Sonic franchise and character are both alive and well.
Jabba, meanwhile, is dead - throttled to death by Leia, using her own chain. Thus he is now but a memory.
What are the names of these two brothers?
Sonic the Hedgehog and Jabba the Hutt.
Additional hint:
Both brothers like to play games
Jabba is a gambler. He's also appeared in Star Wars video games.
Sonic is a video game character, so literally plays games. He has also appeared in "Mario vs Sonic" titles in which the two characters compete in Olympic Games themes mini games.