They are in direct competition in the area they both see as their core strength. Along with all the abstract differences between factions, this is up close and personal. If your company lost out on a trade deal, or a service contract, there is a good bet one of THEM took that money away from you. So there is the constant, but fairly low level effect of being the burr under each others saddle. A constant, nagging, always present irritation.
Which basically twists the knife after the major root cause. Which I feel is, ironically, because they are so similar.
While it's often said that a conquered planet in BTech is a simple change of the flag and things carry on, that can't really be true. The Combine and Capellans are closed, highly controlled rigid societies, complete with secret police. The others are more open. While the Great Houses are more interested in the big fish than hassling every ordinary citizen, for the powerful, rich, elites, there is going to be a big change going from an open to closed society, or the other way around. Each of the other 4 borders means that there will be a large change in how things work for the movers and shakers, change that will be resisted.
Except on the Lyran-Free Worlds border. There it doesn't really seem like it matters. Once you get over the hump of nationalism, a change in border does not mean a transformation in your way of life. Meaning that you will probably adapt to the new regime more quickly that if you had to start learning Bushido, for example, or keeping an eye out for secret police. You are less likely to resent the new regime and pine for the old days.
Which is fine for the average Joe on the street, but must terrify the Archon and Captain-General. It means you are vulnerable to world's flipping quickly. A trickle could turn into a flood. And if it does build momentum, you are probably doomed. Why would worlds wait to be crushed. Why not flip when it is clear who the winner is going to be, and get to keep your company? While the Combine and Confederation would have to take you out world by world as each planet resists in turn, the other side of the Lyran-Free Worlds border simply has to build momentum before becoming an existential threat. And if one of them does manage to eat the other, the resulting Economic and Industrial powerhouse will curb stomp the other houses.
So that Bolan Finger, that isn't just a salient. It's a wedge that could chip off a chunk of the Commonwealth, and if the Free Worlds begin to swarm, the rest might follow. It has to die, and die now, no matter the cost, nuke the hell out of it if you have to, but eliminate that threat. And from the other side, if you can't hold Bolan and it's neighbours, burn it down. In fact, why not give the Lyrans a nice 10 Parsec strip of glass floored, self lighting car parks, to teach them what happens when they mess with us.
Because if you lose a world to the Dracs or Capellans, you'll probably be able to claw it back when the time is right and be welcomed like heroic liberators. Lose it to Steiner/Marik, you probably aren't getting it back. You can't afford to give an inch on this border as it will quickly become light-years.
Which sucks if you happen to live on the border.
And that is why you never turn your back on a Capellan, but shoot Lyrans on sight.