Meillassoux's beautiful prose argues forcefully in favor of the idea that real things can't be self-contradictory. He is afraid that if the law of noncontradiction (LNC) is breached, philosophy opens the door to belief and restrains thinking.
In the first third of After Finitude, Meillassoux rules violations of LNC out of court totally. Then he lets them back in a little bit, via a consideration of paraconsistent logics—that is, logical systems that employ seeming paradoxes but in a relatively constrained way.
Meillassoux constrains their constraint even further by policing paraconsistency—it's supposedly only to do with databases and other software entities.
This slight slippage is in itself interesting—as if Meillassoux can see the danger, and yet as if he's not ready to throw non-LNC thinking out completely. Paging the logic of the supplement...For a philosopher interested in busting through a small crack in the work of Kant (his imagery), this small fissure deserves some attention.
There's more to say on all this.
In any case, for now, the fundamental difference is that I hold that contradictory beings exist—that this is what existence is in some deep sense. In other words, violations of LNC such as the Liar paradox (“This statement is false”) exist as archaeological evidence of something in the ontological realms. The fact that consistent systems are also incomplete (Gödel) is also pretty compelling, despite what Meillassoux says about logical systems and inconsistency.
And on that note, there are plenty of paraconsistent theories that pertain not to software but, for instance, to the way hydrogen atoms behave, and the way waves propagate.
To take a brief example: my concept of the strange stranger, which is roughly the same as the non-rabbit of Brassier's nicely titled essay “Behold the Non-Rabbit.”
What I mean by this is that this rabbit, this one right here, no matter what its name is, no matter what it was or will be (Hegelian dialectics and all that), is inconsistent, in itself.