I could have posted your OP, only mine are 14, 12 and 9!
I think sibling relationships cna be very tricky, and they do/say things that they wouldn't dream of doing/saying to others. Home os the safe zone where they just let out all of those emotions (sigh)
At a guess I would say that the younger one gets frustrated with the older one, who is bigger, faster, stronger etc. He probably nags at and winds the older one up.
The older one probably gets frustrated with the younger one who is an annoying sibling, and he winds him up, just because he can.
I know you have tried a lot of things, and I am struggling myself, so no easy answers, but I wonder if they need things which they can see as theirs. So, older one gets to stay up later on Friday night - Mummy time for the big boy. When I did this with ds we watched Dr Who which was too scary for the younger ones. Focussed time for him, around seomthign he wants to do.
The flip side is give him some responsibility, maybe a household chore to do. Talk about him being the oldest as a positive thing and giving him a status of his own.
Then with younger one, do a similar thing, time for him, he gets to chose, give it name - Jayden's Time, and do some of the stuff that is more appropriate for his age. If you can get the older one on board, you could bribe him with a quite hour on tablet in the other room, while you have special time with younger one. Focus on the differences. Big up things yonger one likes and enjoys.
I guess what I am saying is take away the competativeness by giving them their own things to be good at.