What can you do to compensate for this situation?
What's missing when children don't live in the same house are common experiences -- both positive and negative. Your boys don't have the opportunity to sit and watch a baseball game together on TV or to go out and shoot a few baskets in the afternoons. They don't have as much of a chance to develop a way to relate to one another.
You could try planning activities the family enjoys doing together -- like hikes or trips -- for the times your older son is home. The idea is to create a repertoire of common experiences for your boys. That's not always easy when there is such an age difference but sports or other outdoor activities can bridge that gap.
LET YOUR CHILD TEACH YOU SOMETHING
You want to form an alliance between them, to develop some common area they can enjoy together. Maybe on Fridays, when your ten-year-old comes home from school, his older brother can be a coach for his younger brother's basketball or baseball team.
You can help you older son recognize that he has a vital purpose to serve within the family. I'm not talking so much about chores or household responsibilities here but something that earns him respect like sharing something he's interested in -- be it history or knowledge of computers -- so you as parents become role models of wanting to learn from the older child.
At the same time, your younger son needs that as well so he doesn't feel that during the time his brother is away he's important but as soon as the older one comes home, he's nobody. He's also old enough for you to be able to sit him down and explain to him that his older brother needs extra attention because he's not home much.
It would help if each brother understood that the other one is also vulnerable. The older one thinks the younger one is home all the time and gets all the attention and parenting and the younger one feels that as soon as older one walks in, "He's king and I'm dispensable".
If each could be helped to articulate his feelings to the other and to his parents, then each one can recognize the other's vulnerabilities and feel less threatened and maybe even more empathic.
Other than that, have a lot of company over for weekend meals, so they have less of a chance to fight!