If you think about it, every single Kyurem-B set is a wallbreaker. They all run pretty similar moves -- they just use different means to get to that end. Technically if you want you could have three Kyurem-B sets as you said -- special attacker, physical attacker, and utility check -- but that wouldn't cover the whole story. The difference between Kyurem-B sets is all nuance, minor differences here and there -- hell physical isn't even that different from special. The question that arises when you have such similar sets is where to separate them and allow for this nuance.
We've agreed to disagree on Earth Power being compulsory on LO (over IRC) -- I'll just assume that it's mandatory. That's one difference between the sets. Also, there's a nature difference. But, neither of those are really huge, and I see exactly what you're getting at when you're saying both could easily be covered by a slash.
The major difference is that LO has special attacking prowess, whilst CB does not. Yes, both wallbreak, but LO utilizes special coverage to do so in addition to its physical hard-hitting attacks. Ice Beam is basically filler on CB -- it doesn't really count as a mixed attacker. There's nothing else remotely useful in Kyurem-B's movepool, so it runs Ice Beam. You'll use it once in a blue moon, while Life Orb will use it liberally, as it easily fells foes such as Trevenant, Celebi, and Mega Venusaur. Earth Power provides another dimension of coverage, and helps against Rotom-W, Heatran, and Bisharp. What can they do when you 2HKO with Fusion Bolt? First of all, if you've already killed something, they can burn you. About burn: CB is way more crippled by it than LO. Yes, LO doesn't want to get burned, but it's not the end of the world if it does, as it still has Ice Beam and Earth Power to fall back on. CB meanwhile is entirely crippled by a burn -- made useless. This is another fundamental difference between the two. CB is pretty much 100% physical (as in you'll be using physical moves almost always), while LO is truly mixed.
Also, with the ability to switch moves, LO is able to take out an array of threats more easily. CB meanwhile is reliant on Outrage to guarantee it takes out the opposing Pokemon, and against Fairy-types can't even do that much. Dragon Claw and Fusion Bolt simply aren't strong enough to smash apart resists like Outrage can. With its coverage and ability to switch moves, LO can just pick the appropriate coverage move instead of resorting to Outrage in many cases. Fundamental difference #3.
It's also easier to force out CB Kyurem-B. If a good resist/immunity to an attack switches in, it must switch out. Meanwhile, with LO, you can switch to the appropriate attack and murder the foe. CB cannot boast this versatility in move selection. Fundamental difference #4.
This is not to say CB Kyurem-B is a pile of garbage. Its physical power is absolutely ridiculous. If you want something dead ASAP, CB Kyurem-B is your man. It's just that powerful. But, it's not like LO for the reasons I've mentioned.
I've mention four fundamental differences. If you look at the bigger picture, yes, both are wallbreakers and are EV'd to attack physically. They run similar moves bar the one in the last slot, and the natures can be slashed together. But, as I've stated, LO Kyurem-B is different in various crucial aspects from CB. It's not like Keldeo. CB Kyurem-B is trading the ability to attack specially and utilize its excellent coverage to the fullest extent, whereas Keldeo just trades the ability to switch moves. CB Kyurem-B is also utilizing a lock-in move as its big puncher STAB; Keldeo is not. Keldeo can just switch out should a situation go awry, but Kyurem-B does not have that luxury when it inevitably uses Outrage to nab a kill it needs. The scenarios are different.