What are the current explanations and models for the formation of a central bar in some galaxies, and what is the difference that can produce a barred galaxy instead of a spiral one ?
It's my impression from skimming the literature just now that much of the current activity in this field is based on N-body simulations. It's hard for me as an outsider to evaluate the analytical scenarios that are being tested, but I think that the key idea is that certain Lindblad resonances which excite density waves are the main instabilities that lead to bar formation. See also section (4) of this page, which gives a broad overview of stellar dynamics in galaxies.
In short, there are several guesses out there. See this paper by Steinmetz for an overview of the understanding of galaxy shapes in general in 2002.
The key citation in the wikipedia article linked to by Cem is a 2002 paper by Bournard and Combes (free on the arxiv) which describes gravitational N-body simulations which follow processes of bar formation and re-formation. They find a strong dependence on the frequency of bar formation on the distribution of mass in the galaxy:
Another paper from 2002 by Steinmetz and Navarro does a different simulation and observes that a bar is formed in their model galaxy due to the tidal forces from a satellite galaxy. Here's a related image from their paper: