|Relaxed vs. hard-nosed would be one way of characterizing the differences. Objectivist guerilla warfare would be another.|
Basically, ARI is the official Objectivist organization, founded by Leonard Peikoff, Rand’s supposed “intellectual heir.”
TOC was started in 1990 by David Kelley shortly after Peikoff made him persona non grata for writing a paper entitled “A Question of Sanction,” which defended Kelley’s decision to speak to a libertarian group. The key issue involved was whether or not the Objectivist ethics is consistent with a policy of intellectual tolerance. Peikoff’s essay in defense of the excommunication is entitled “Fact and Value.” Kelley later published a detailed response to Peikoff entitled Truth and Toleration (subsequently revised under the title of The Contested Legacy of Ayn Rand.) You can read some of the seminal documents here.
This was the second great Objectivist schism. Peikoff tried to make the case that the underlying philosophical origin was identical to Ayn Rand’s repudiation of Nathaniel Branden in 1968, but no one outside Peikoff’s sphere of influence is likely to agree.
The official policy of ARI toward TOC is utter and unqualified contempt. The unofficial policy of TOC toward ARI is that Peikoff is a misguided authoritarian tyrant whose followers will hopefully see the light some day.
As it stands today, ARI’s intellectual leadership does a good job of publishing books and articles which clearly articulate the application of Objectivist principles, but manage to make the movement look like a lunatic fringe by impugning the honesty of anyone who expresses significant disagreement. TOC’s leadership demonstrates a grasp of the proper approach to spreading a new philosophy—i.e., they encourage debate on the assumption that you cannot shame people into agreement and the more rational position will win out—but they often display a disturbingly cavalier attitude toward philosophical consistency and fundamental principles in their published offerings. (See the thread “TOC Watch” in the General Forum section.)
Hope this answers your question, without pouring cold water on your enthusiasm for the ideas. It is, after all, the ideas that really matter here. Alas, Objectivism’s superior alliance with reality has not spared it from the same bitter infighting that has characterized all major philosophical movements throughout human history.