Pretty much every well-disciplined modern military (which is what the militarized Spartans model themselves after) advocates rigourous improvement of the self with devotion to the whole and the corps above all.
Where the Spartan subchapter would splinter and break with the orthodoxy per Dioxine is in their emphasis on militarism, sacrifice and conflict. Conservative survivalists set on isolation vs aggressive militarists set on conquest is a pretty substantial divide; Randian/individualist elements don't have to enter into it in order to create a substantial schism: the survivalists raid occasionally and strictly as necessary, the militarists do it often as a rite of passage to purge the weak/unworthy and a mode of honing their skills. Even outside of the latter's rituals which the former undoubtedly finds wasteful and unnecessary, there's plenty to drive them apart. That said, the two types of Spartans may not necessarily be driven to hate each other, and might even work together every now and then, though I can see an obvious mutual contempt forming: militarists think the survivalists are weak cowards, and the survivalists look at the militarists as reckless fools, with each feeling they embody the 'true' Spartan creed.
A prime example of such a divide would be the Brotherhood of Steel: the traditional chapter seeking conservation of the old ways and isolationism vs the splinter that wanted to expand (including through aggressive militarism/exchanging protection and policing for supplies and soldiers) by tapping the wasteland for personnel (ala Fallout 3, Fallout Tactics).
