An-Synd and An-Com are the same in that they both support greater worker control over the means of production, but they are slightly different in the way that it is implemented. An-Coms generally believe in communism achieved through violent revolution, while An-Synds believe in more of a militarized sort of "labor union" force to meet and act directly in worker solidarity that eventually grows large enough to be able to completely disrupt the capitalist system by going on strike (essentially, enough people are organized and then act directly through strike to force business to close). The next step is to entirely seize the means of production for workers and dissolve the state in favor of a sort of mass meeting/council of workers who can act without bosses or leaders directing them on what to do. So An-Synds and An-Coms have the same end goal, just different means of reaching that end goal.
To answer your question about the difference between the state and decentralized political organizations, I guess I just need to further evaluate on what I said above. Rather than having groups of representatives or bureaucrats assembling to make the decisions for everyone else or a set state body governing everything, anarchist political organization advocates for direct, individual action. Essentially, an accumulation of workers in a town hall that speak with each other rather than speaking to bureaucrats.
As for capitalism and socialism, separating individuals as either definitively socialist or definitively capitalist is dealing with absolutes, which is quite the reactionary way of thinking. There is quite a broad spectrum, and I personally find it silly that people try to definitively place others as capitalist or socialist with no recognition for the wide spectrum. Even within socialism itself there's a massive spectrum of different ideologies.
I hope I could help! However, I may not be entirely correct on my statements or I may not have described it in the best way, as I am not an anarchist myself, so please correct or further evaluate my statements if you feel the need to, anyone!



